#8095833

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	#45680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

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

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, 8003029);