#411004

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

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, 318200);