#7203532

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^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,e^c=ei,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, 7110728);