#7500484

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1208 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)(33,39)(36,48)(37,45)(40,52)(41,46)(42,49)(43,50)(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)(25,27,31,30)(26,32,35,44)(28,39,41,36)(29,42,43,37)(33,46,48,38)(34,49,50,45)(40,51,54,53)(47,52,56,55);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,30,39,31,41,27,36)(26,33,32,38,35,48,44,46)(29,40,42,53,43,54,37,51)(34,47,45,52,50,56,49,55);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29,31,43)(26,34,35,50)(27,37,30,42)(28,40,41,54)(32,45,44,49)(33,47,48,56)(36,51,39,53)(38,52,46,55);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,27,31,30)(26,32,35,44)(28,36,41,39)(29,42,43,37)(33,38,48,46)(34,49,50,45)(40,53,54,51)(47,55,56,52)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407680);