#7035200

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,38)(28,43)(29,35)(30,36)(31,37)(32,47)(33,41)(34,44)(39,52)(40,54)(42,50)(45,49)(46,51)(48,56)(53,55);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,48)(35,52)(36,50)(37,38)(43,49)(44,55)(46,56)(47,54)(51,53);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,29)(26,34,37,51)(27,39,41,45)(30,44,32,46)(33,48,38,53)(35,47,43,36)(40,55,42,56)(49,50,52,54);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,28)(26,35,37,43)(27,39,41,45)(30,44,32,46)(33,49,38,52)(34,47,51,36)(40,56,42,55)(48,50,53,54);(10,16)(11,18)(15,24)(17,19)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,44,29,46)(33,50,38,54)(34,35,51,43)(39,55,45,56)(48,52,53,49)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=i,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, 6942396);