#6954011

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(25,26,30,35)(27,31,37,44)(28,33,39,47)(29,32,41,34)(36,43,50,55)(38,42,51,45)(40,46,52,48)(49,53,56,54);(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)(28,36)(29,38)(30,37)(32,42)(33,43)(34,45)(35,44)(39,50)(40,49)(41,51)(46,53)(47,55)(48,54)(52,56);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,32)(29,41)(31,42)(33,46)(34,35)(38,51)(40,52)(43,53)(44,45)(47,48)(49,56)(54,55);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28,30,39)(26,33,35,47)(27,36,37,50)(29,40,41,52)(31,43,44,55)(32,46,34,48)(38,49,51,56)(42,53,45,54);(10,14)(15,21)(16,22)(23,24)(27,37)(28,39)(31,44)(33,47)(38,51)(40,52)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,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, 6861207);