#6954432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fhi,d^2=fhi,e^2=i,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=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6861628);