#6954105

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(27,31,36,45)(28,38,39,48)(29,41,43,33)(32,42,47,40)(35,49,50,56)(37,53,52,46)(44,54,55,51);(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)(28,35)(29,37)(30,36)(32,44)(33,46)(34,45)(38,49)(39,50)(40,51)(41,53)(42,54)(43,52)(47,55)(48,56);(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)(29,42)(30,39)(31,44)(33,48)(34,47)(36,50)(37,54)(38,41)(40,43)(45,55)(46,56)(49,53)(51,52);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,36)(31,45)(35,50)(37,52)(44,55)(46,53)(49,56)(51,54);(10,14)(15,21)(16,22)(23,24)(28,39)(32,47)(35,50)(38,48)(40,42)(44,55)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=fh,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, 6861301);