#8464096

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	#42547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=g,e^2=i,f^2=gi,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, 8371292);