#9937804

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

3954 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=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, 9845000);