#6776120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1171 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6683316);