#6794662

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

1206 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)(12,20,21,15)(18,22,24,23)(25,26,31,37,32,38,30,36)(27,33,40,49,41,50,46,53)(28,42,44,52,29,35,45,34)(39,48,55,51,43,56,54,47);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,32,41)(26,33,38,50)(28,43,29,39)(30,46,31,40)(34,51,52,47)(35,48,42,56)(36,53,37,49)(44,54,45,55);(2,5)(6,7)(25,28,32,29)(26,34,38,52)(27,39,41,43)(30,44,31,45)(33,47,50,51)(35,36,42,37)(40,54,46,55)(48,49,56,53);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,32,28)(26,35,38,42)(27,39,41,43)(30,44,31,45)(33,48,50,56)(34,36,52,37)(40,55,46,54)(47,49,51,53);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(22,23)(25,30,32,31)(26,36,38,37)(27,40,41,46)(28,44,29,45)(33,49,50,53)(34,42,52,35)(39,54,43,55)(47,56,51,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6701858);