#8317516

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

4075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8224712);