#8471130

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gi,e^2=gi,f^2=hi,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, 8378326);