#7500551

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=gi,d^2=fh,e^2=h,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=dh,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, 7407747);