#6937128

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6844324);