#7036719

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,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=eh,d^c=dh,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, 6943915);