#328448

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
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

883 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,21)(20,23)(22,24)(25,26,30,33)(27,34,36,40)(28,32,29,37)(31,38,39,35);(2,5)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,22,23,21)(28,29)(31,39)(32,37)(35,38);(3,7)(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,27,30,36)(26,31,33,39)(28,35,29,38)(32,40,37,34);(1,3,4,7)(2,6,5,8)(12,20)(15,23)(18,24)(21,22)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,40,39,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235644);