#235495

order	512
nilpotency class	3
isoclinism	isoclinic class of #235492
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142691);