#6789379

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1174 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)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,30,35)(27,36,31,32)(28,39,42,47)(29,43,45,34)(33,37,50,40)(38,48,46,52)(41,56,44,55)(49,54,51,53);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,45,46)(33,39,50,47)(34,48,43,52)(41,54,44,53)(49,55,51,56);(2,5)(6,8)(25,28,27,37,30,42,31,40)(26,33,32,47,35,50,36,39)(29,44,38,54,45,41,46,53)(34,51,48,55,43,49,52,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,30,45)(26,34,35,43)(27,38,31,46)(28,41,42,44)(32,48,36,52)(33,49,50,51)(37,53,40,54)(39,55,47,56);(2,5)(6,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(28,42)(29,45)(33,50)(34,43)(37,40)(38,46)(39,47)(48,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fhi,e^2=fi,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, 6696575);