#232735

order	512
nilpotency class	3
isoclinism	isoclinic class of #232724
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1109 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1109 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)(3,13,17,23)(4,9,20,28)(5,10,6,11)(8,16,24,15)(14,29,30,25)(18,22,31,32)(19,26,21,27)(33,34)(35,40)(36,47)(37,42)(38,51)(39,43)(41,48)(44,60)(45,54)(46,56)(49,58)(50,55)(52,59)(53,63)(57,61)(62,64);(1,3,7,17)(2,8,12,24)(4,18,20,31)(5,16,6,15)(9,25,28,29)(10,13,11,23)(14,21,30,19)(22,27,32,26)(33,35)(34,40)(36,44)(37,50)(38,45)(39,46)(41,57)(42,55)(43,56)(47,64)(48,63)(49,62)(51,54)(52,61)(53,59)(58,60);(1,4,7,20)(2,9,12,28)(3,14,17,30)(5,19,6,21)(8,22,24,32)(10,26,11,27)(13,29,23,25)(15,18,16,31)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,58)(43,59)(45,61)(46,62)(47,51)(50,63)(54,60)(55,64)(56,57);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,26,28,27)(14,18,30,31)(22,25,32,29)(33,37,39,38)(34,42,43,51)(35,45,46,50)(36,48,49,52)(40,54,56,55)(41,58,59,47)(44,61,62,63)(53,60,57,64)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fi,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, 139931);