#7227576

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1245 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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, 7134772);