#6964485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(15,22)(20,23)(21,24)(25,26,29,34)(27,36,37,31)(28,40,41,33)(30,32,43,35)(38,45,50,49)(39,44,51,46)(42,48,53,47)(52,55,56,54);(2,5)(3,7)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27)(26,31)(28,38)(29,37)(30,39)(32,46)(33,45)(34,36)(35,44)(40,49)(41,50)(42,52)(43,51)(47,55)(48,54)(53,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,50)(40,48)(42,53)(45,54)(49,55);(2,5)(6,8)(10,17)(12,18)(15,19)(20,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,45)(32,47)(34,40)(35,48)(36,49)(37,50)(39,52)(43,53)(44,54)(46,55)(51,56);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(26,34)(28,41)(31,36)(32,35)(38,50)(42,53)(44,46)(52,56)

Polycyclic presentation:

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

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, 6871681);