#6847961

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=f,e^2=h,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, 6755157);