#7500027

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

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,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=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, 7407223);