#6634854

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542050);