#349092

order	512
nilpotency class	3
isoclinism	isoclinic class of #349070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=di,f^c=fi,g^d=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, 256288);