#6855832

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

1170 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6763028);