#6954100

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(27,36)(28,40)(29,35)(30,34)(31,45)(32,38)(33,43)(37,52)(39,48)(41,50)(42,49)(44,47)(46,55)(51,53)(54,56);(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)(26,32)(28,37)(29,44)(30,38)(31,39)(33,46)(34,36)(35,48)(40,56)(41,54)(42,55)(43,53)(45,47)(49,51)(50,52);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,41)(26,33,34,49)(27,37,38,54)(29,43,31,42)(32,46,36,51)(35,50,45,40)(39,55,44,53)(47,56,48,52);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,34)(27,38)(28,41)(35,45)(39,44)(42,43)(46,51)(52,56);(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,34,45)(27,39,38,44)(28,42,41,43)(32,47,36,48)(33,40,49,50)(37,53,54,55)(46,52,51,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,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

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, 6861296);