#7232118

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7139314);