#6765794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1318 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,36)(28,41)(29,35)(30,32)(31,46)(33,40)(34,42)(37,53)(38,54)(39,47)(43,52)(44,51)(45,50)(48,56)(49,55);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(26,32)(27,37)(28,42)(35,46)(39,40)(44,45)(48,49)(52,54);(2,6)(3,7)(25,27,29,39,30,37,31,40)(26,33,46,53,32,47,35,36)(28,43,45,55,42,38,44,56)(34,49,50,52,41,48,51,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28)(26,34)(27,38)(29,45)(30,42)(31,44)(32,41)(33,48)(35,51)(36,52)(37,43)(39,56)(40,55)(46,50)(47,49)(53,54);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,29,30,31)(26,35,32,46)(27,39,37,40)(28,44,42,45)(33,36,47,53)(34,50,41,51)(38,55,43,56)(48,54,49,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=gh,d^2=fi,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,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6672990);