#8195382

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8102578);