#328428

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

816 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)(25,26,29,33)(27,35,28,31)(30,39,38,34)(32,36,40,37);(2,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,29,28)(26,31,33,35)(30,37,38,36)(32,39,40,34);(3,7)(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28,29,27)(26,32,33,40)(30,36,38,37)(31,34,35,39);(1,3,4,7)(2,6,5,8)(12,20)(15,23)(18,24)(21,22)(27,28)(31,35)(32,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 235624);