#7277979

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1176 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,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, 7185175);