#10424609

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

4025 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(2,4)(7,9)(8,12)(15,16)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,f^2=gh,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, 10331805);