#8464254

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=gi,f^2=i,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, 8371450);