#424332

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331528);