#9814637

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

3952 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34)(35,37)(36,41)(38,47)(39,40)(42,44)(43,46)(45,48);(2,7)(4,8)(9,15)(12,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,35)(34,37)(36,44)(38,45)(39,43)(40,46)(41,42)(47,48);(18,23)(22,25)(24,31)(30,32)(34,40)(35,43)(36,45)(42,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,36,39,45)(34,41,40,48)(35,44,43,38)(37,42,46,47);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(33,37)(34,35)(36,42)(38,48)(39,46)(40,43)(41,44)(45,47);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,38,39,44)(34,42,40,47)(35,45,43,36)(37,41,46,48)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9721833);