#7277986

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

1178 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)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,48)(39,51)(40,45)(42,46)(43,50)(44,49)(47,53)(52,56)(54,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(26,32)(27,38)(28,42)(35,36)(40,41)(43,44)(47,55)(51,56);(2,5)(3,7)(25,27,30,38)(26,33,32,45)(28,39,42,52)(29,41,31,40)(34,47,46,55)(35,37,36,48)(43,54,44,53)(49,56,50,51);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,34)(27,39)(29,44)(30,42)(31,43)(32,46)(33,47)(35,50)(36,49)(37,51)(38,52)(40,54)(41,53)(45,55)(48,56);(2,5)(6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(39,53,52,54)(47,51,55,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=hi,e^2=f,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, 7185182);