#237094

order	512
nilpotency class	3
isoclinism	isoclinic class of #237079
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

1102 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144290);