#9835835

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
prime radical	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=ghi,e^2=gh,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, 9743031);