#8093218

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	#46106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

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

Polycyclic presentation:

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

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, 8000414);