#7129352

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=h,e^2=fi,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, 7036548);