#7500543

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1233 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,5)(3,6,7,8)(10,14)(12,20)(15,17)(19,24)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,41)(36,47)(37,45)(39,50)(40,51)(42,48)(43,54)(46,52)(49,55)(53,56);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,31,30)(26,32,35,44)(28,36,40,41)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,53,52)(46,50,55,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,27,36,31,40,30,41)(26,33,44,51,35,47,32,38)(29,39,37,49,42,53,43,52)(34,46,54,56,48,55,45,50);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29)(26,34)(27,37)(28,39)(30,43)(31,42)(32,45)(33,46)(35,48)(36,49)(38,50)(40,53)(41,52)(44,54)(47,55)(51,56);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(28,40)(29,42)(33,47)(34,48)(36,41)(37,43)(38,51)(45,54)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,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, 7407739);