#300416

order	512
nilpotency class	3
isoclinism	isoclinic class of #300397
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fh,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,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207612);