#7500039

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

Polycyclic presentation:

2^9|a^2=fi,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=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, 7407235);