#6788717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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

1177 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,36,45)(28,38,40,55)(29,42,43,33)(32,47,49,41)(35,50,39,54)(37,53,52,46)(44,56,48,51);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,46)(34,45)(35,40)(38,54)(41,56)(42,53)(43,52)(44,49)(47,51)(50,55);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,29,41,30,40,43,47)(26,32,33,38,34,49,42,55)(27,35,37,51,36,39,52,56)(31,44,46,50,45,48,53,54);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(27,36)(28,40)(31,45)(32,49)(37,52)(38,55)(41,47)(46,53);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,36)(28,40)(31,45)(32,49)(37,52)(38,55)(41,47)(46,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=hi,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=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6695913);