#10420400

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

4264 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10327596);