#294424

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

893 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23)(25,26,30,33)(27,34,35,40)(28,32,29,38)(31,36,39,37);(2,5)(6,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,17)(18,21)(23,24)(27,35)(28,29)(32,38)(34,40);(9,12)(10,15)(11,18,19,20)(13,21)(14,22,17,24)(16,23)(25,27,28,36,30,35,29,37)(26,31,38,40,33,39,32,34);(1,3,4,8)(2,6,5,7)(10,16)(14,17)(15,23)(22,24)(25,28,30,29)(26,32,33,38)(27,36,35,37)(31,34,39,40)

Polycyclic presentation:

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

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, 201620);