#300450

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62435	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207646);