#7053305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6960501);