#8533642

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440838);