#409612

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 316808);