#9835840

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(17,18,22,24)(19,27,29,23)(20,25,28,32)(21,30,31,26)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19,22,29)(18,23,24,27)(20,21,28,31)(25,26,32,30)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,48)(42,47);(2,8)(3,10)(9,13)(11,16)(18,24)(23,27)(25,32)(26,30)(33,34,39,37)(35,40,46,43)(36,41,45,47)(38,42,44,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,48,43,42);(1,4,6,7)(2,3,8,10)(5,14,15,12)(9,16,13,11)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31)(33,37,39,34)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,5,6,15)(2,9,8,13)(3,11,10,16)(4,12,7,14)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,47,43,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,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, 9743036);