#7500555

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=g,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, 7407751);