#459759

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16348
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,g^d=gi

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, 366955);