#8188232

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
prime radical	#41825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095428);