#244622

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 937 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151818);