#7500610

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1229 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)(30,45)(31,35)(33,40)(36,49)(37,46)(39,55)(41,48)(42,56)(43,51)(44,52)(47,54)(50,53);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,37,44,43)(33,38,49,48)(34,46,52,51)(39,53,42,54)(47,55,50,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,27,36,31,41,30,40)(26,33,45,48,35,49,32,38)(29,42,37,54,44,39,43,53)(34,50,51,55,52,47,46,56);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29,31,44)(26,34,35,52)(27,37,30,43)(28,39,41,42)(32,46,45,51)(33,47,49,50)(36,53,40,54)(38,55,48,56);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,27,31,30)(26,32,35,45)(28,40,41,36)(29,43,44,37)(33,48,49,38)(34,51,52,46)(39,53,42,54)(47,55,50,56)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=gi,d^2=fhi,e^2=hi,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, 7407806);