#6770539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6677735);