#10534322

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

3989 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)(3,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18,24,26)(19,27,22,23)(20,29,21,25)(28,31,30,32)(33,34,39,43)(35,40,45,37)(36,41,38,42)(44,48,46,47);(17,19,24,22)(18,23,26,27)(20,28,21,30)(25,31,29,32)(33,35,39,45)(34,40,43,37)(36,46,38,44)(41,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,24,21)(18,25,26,29)(19,28,22,30)(23,31,27,32)(33,36,39,38)(34,41,43,42)(35,44,45,46)(37,47,40,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,24,20)(18,25,26,29)(19,28,22,30)(23,32,27,31)(33,34)(35,37)(36,41)(38,42)(39,43)(40,45)(44,47)(46,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,22,24,19)(18,23,26,27)(20,30,21,28)(25,31,29,32)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,46)(43,45);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,23)(18,22)(19,26)(20,31)(21,32)(24,27)(25,30)(28,29)(33,38,39,36)(34,42,43,41)(35,44,45,46)(37,47,40,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10441518);