#232498

order	512
nilpotency class	3
isoclinism	isoclinic class of #232487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1101 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1101 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,17,24)(4,18,22,27)(5,10,6,11)(8,16,25,15)(9,20,28,21)(14,30,31,23)(19,29,32,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,50)(44,51)(45,52)(46,54)(47,55)(48,56)(49,57)(53,59)(58,62)(60,63)(61,64);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,13)(11,24)(12,25)(14,20)(18,23)(21,31)(22,32)(27,30)(28,29)(33,35,38,45)(34,39,42,52)(36,43,47,59)(37,48,49,44)(40,53,55,50)(41,56,57,51)(46,60,61,58)(54,62,64,63);(1,4,7,22)(2,9,12,28)(3,14,17,31)(5,20,6,21)(8,23,25,30)(10,18,11,27)(13,29,24,26)(15,19,16,32)(33,36)(34,40)(35,43)(37,46)(38,47)(39,50)(41,54)(42,55)(44,58)(45,59)(48,60)(49,61)(51,62)(52,53)(56,63)(57,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,20,22,21)(8,24,25,13)(9,18,28,27)(14,19,31,32)(23,26,30,29)(33,37,38,49)(34,41,42,57)(35,44,45,48)(36,46,47,61)(39,51,52,56)(40,54,55,64)(43,58,59,60)(50,62,53,63)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139694);