#7293176

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234453
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=gi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7200372);