#8436226

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

4024 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343422);