#409750

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,27,15,12,25,18)(11,19,23,20,24,16,22,31)(13,17,28,39,29,26,38,32)(21,33,36,34,37,30,35,40);(1,3,4,7)(2,5,6,8)(9,11,14,22,15,24,25,23)(10,16,27,20,12,19,18,31)(13,21,28,35,29,37,38,36)(17,30,39,34,26,33,32,40);(9,12)(10,15)(11,20)(13,26)(14,18)(16,23)(17,29)(19,22)(21,34)(24,31)(25,27)(28,32)(30,36)(33,35)(37,40)(38,39);(2,6)(5,8)(9,13,14,28,15,29,25,38)(10,17,27,39,12,26,18,32)(11,21,22,35,24,37,23,36)(16,30,20,34,19,33,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316946);