#6790894

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1229 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,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, 6698090);