#10247624

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

3995 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,40,44)(35,45,38,41)(36,42,37,46)(39,43,48,47);(2,3)(4,11)(8,10)(12,13)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35,40,38)(34,41,44,45)(36,43,37,47)(39,46,48,42);(2,3)(7,9)(8,10)(15,16)(18,25)(19,29)(20,31)(27,32)(33,36,40,37)(34,42,44,46)(35,43,38,47)(39,45,48,41);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,37,40,36)(34,42,44,46)(35,43,38,47)(39,41,48,45);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,38,40,35)(34,41,44,45)(36,47,37,43)(39,46,48,42);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,39)(34,43)(35,46)(36,45)(37,41)(38,42)(40,48)(44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154820);