#324112

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=fh,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231308);