#6776058

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

1175 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)(25,26,31,36)(27,32,39,49)(28,40,29,34)(30,44,45,35)(33,43,50,41)(37,47,42,51)(38,53,55,48)(46,54,56,52);(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)(26,32)(28,37)(29,42)(30,38)(31,39)(33,46)(34,51)(35,48)(36,49)(40,47)(41,52)(43,54)(44,53)(45,55)(50,56);(2,5)(6,8)(25,28,31,29)(26,33,36,50)(27,37,39,42)(30,43,45,41)(32,46,49,56)(34,44,40,35)(38,54,55,52)(47,48,51,53);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,31,28)(26,34,36,40)(27,37,39,42)(30,43,45,41)(32,47,49,51)(33,44,50,35)(38,52,55,54)(46,48,56,53);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,29)(33,50)(34,40)(37,42)(41,43)(46,56)(47,51)(52,54)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,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,e^c=ei,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, 6683254);