#7500625

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	#32668	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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)(27,30)(28,36)(29,33)(31,34)(32,37)(35,38)(39,40);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27)(26,30)(28,36)(29,34)(31,33)(32,39)(35,38)(37,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,29,39,31,38,30,37)(26,32,27,35,34,40,33,36);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,26)(27,29)(28,32)(30,33)(31,34)(35,39)(36,37)(38,40);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,31,30)(26,33,34,27)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=gi,d^2=fhi,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=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, 7407821);