#300418

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
prime radical	#62436	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1033 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)(12,19)(13,31)(14,21)(15,20)(16,22)(17,25)(18,29)(24,32)(26,35)(27,34)(28,38)(30,37)(33,40)(36,39);(1,3,7,2)(4,5,8,6)(9,11,15,26,16,27,14,25)(10,17,21,34,22,35,20,23)(12,28,13,24,30,39,29,40)(18,36,37,32,31,38,19,33);(1,4,5,3,7,8,6,2)(9,12,14,29,16,30,15,13)(10,18,20,19,22,31,21,37)(11,24,26,39,27,40,25,28)(17,32,34,38,35,33,23,36);(9,13,15,30,16,29,14,12)(10,19,21,18,22,37,20,31)(11,24,26,39,27,40,25,28)(17,33,34,36,35,32,23,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=f,d^2=eh,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, 207614);