#6755902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(15,21)(20,24)(22,23)(25,26,30,36)(27,37,38,32)(28,42,45,34)(29,35,31,33)(39,48,43,51)(40,49,41,47)(44,52,46,50)(53,56,54,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27)(26,32)(28,43)(29,40)(30,38)(31,41)(33,49)(34,51)(35,47)(36,37)(39,45)(42,48)(44,54)(46,53)(50,55)(52,56);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,33)(27,38)(29,31)(32,47)(34,50)(35,36)(37,49)(39,43)(42,52)(44,46)(48,55)(51,56);(10,17)(15,19)(16,18)(23,24)(25,28,30,45)(26,34,36,42)(27,39,38,43)(29,46,31,44)(32,48,37,51)(33,50,35,52)(40,54,41,53)(47,55,49,56);(2,6)(5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,36,33)(27,40,38,41)(28,44,45,46)(32,47,37,49)(34,50,42,52)(39,53,43,54)(48,56,51,55)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6663098);