#7223750

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1233 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fh,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=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7130946);