#300419

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	#62437	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1001 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)(9,10,16,22)(11,23,28,34)(12,19,31,37)(13,18,30,38)(14,21,15,20)(17,27,35,26)(24,33,25,32)(29,36,40,39);(1,3,7,2)(4,5,8,6)(9,11,15,27,16,28,14,26)(10,17,21,34,22,35,20,23)(12,29,13,25,31,40,30,24)(18,36,19,33,38,39,37,32);(1,4,5,3,7,8,6,2)(9,12,14,30,16,31,15,13)(10,18,20,37,22,38,21,19)(11,24,27,29,28,25,26,40)(17,32,34,36,35,33,23,39);(9,13,15,31,16,30,14,12)(10,19,21,38,22,37,20,18)(11,25,27,40,28,24,26,29)(17,33,34,39,35,32,23,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fh,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, 207615);