#414982

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15181
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1209 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322178);