#6954111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,30,34)(27,31,36,46)(28,38,40,50)(29,42,44,33)(32,43,49,41)(35,51,39,56)(37,54,53,47)(45,55,48,52);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,47)(34,46)(35,40)(38,56)(41,55)(42,54)(43,52)(44,53)(45,49)(50,51);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,32)(27,35)(29,43)(30,40)(31,45)(33,50)(34,49)(36,39)(37,55)(38,42)(41,44)(46,48)(47,56)(51,54)(52,53);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,36)(28,40)(31,46)(32,49)(37,53)(38,50)(41,43)(47,54);(10,14)(15,21)(16,22)(23,24)(27,36)(28,40)(31,46)(32,49)(37,53)(38,50)(41,43)(47,54)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=fhi,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

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, 6861307);