#6777046

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6684242);