#331129

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

710 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)(3,6)(4,5)(7,8)(9,10,13,16,14,17,12,21)(11,18,19,24,20,23,22,15)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(2,5)(6,8)(25,27,29,35,28,34,36,33)(26,30,32,37,31,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(3,7)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238325);