#7036638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6943834);