#7500038

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=gi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,e^d=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, 7407234);