#10440033

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

4124 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(2,7)(4,9)(8,15)(12,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,3,6,11)(2,4,7,9)(5,13,14,10)(8,16,15,12)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=i,f^2=h,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, 10347229);