#6489134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6396330);