#8187014

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

4001 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=ghi,e^2=h,f^2=g,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, 8094210);