#8096877

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=g,e^2=gi,f^2=gi,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, 8004073);