#324392

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

598 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)(11,17,19,24)(13,21,18,16)(14,22,23,20)(25,26)(27,31)(28,38)(29,34)(30,33)(32,35)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24)(25,27,35,36,30,28,29,37)(26,31,32,40,33,38,34,39);(9,11,18,22,12,19,13,20)(10,14,21,24,15,23,16,17)(25,28,30,27)(26,31,33,38)(29,37,35,36)(32,39,34,40);(2,6)(5,7)(10,15)(14,23)(16,21)(17,24)(25,29,30,35)(26,32,33,34)(27,36,28,37)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=fh,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,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, 231588);