#6702715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,36)(28,41)(29,35)(30,33)(31,34)(32,40)(37,46)(38,47)(39,53)(42,52)(43,56)(44,49)(45,50)(48,55)(51,54);(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,30,40)(26,32,33,36)(28,38,31,37)(29,43,44,54)(34,47,41,46)(35,51,49,56)(39,42,55,45)(48,52,53,50);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(26,33)(27,37)(28,31)(32,46)(35,49)(36,47)(38,40)(39,54)(42,45)(43,55)(48,51)(53,56);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,30,31)(26,34,33,41)(27,38,40,37)(29,42,44,45)(32,47,36,46)(35,50,49,52)(39,43,55,54)(48,56,53,51);(11,21)(15,18)(19,20)(23,24)(25,29,30,44)(26,35,33,49)(27,39,40,55)(28,42,31,45)(32,48,36,53)(34,50,41,52)(37,54,38,43)(46,51,47,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6609911);