#9835842

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3893 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)(19,26,30,23)(20,29,31,25)(21,28,32,27)(33,34,39,37)(35,43,44,40)(36,45,46,41)(38,42,48,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,35,39,44)(34,40,37,43)(36,42,46,47)(38,41,48,45);(2,8)(3,10)(9,13)(11,16)(18,24)(20,31)(23,26)(27,28)(33,34,39,37)(35,40,44,43)(36,41,46,45)(38,42,48,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20)(18,25)(19,28)(21,26)(22,31)(23,32)(24,29)(27,30)(33,36,39,46)(34,41,37,45)(35,42,44,47)(38,40,48,43);(1,4,6,7)(2,3,8,10)(5,14,15,12)(9,16,13,11)(17,19)(18,26)(20,28)(21,25)(22,30)(23,24)(27,31)(29,32)(33,37,39,34)(35,40,44,43)(36,45,46,41)(38,42,48,47);(1,5,6,15)(2,9,8,13)(3,11,10,16)(4,12,7,14)(17,21)(18,27)(19,29)(20,26)(22,32)(23,31)(24,28)(25,30)(33,38,39,48)(34,42,37,47)(35,41,44,45)(36,40,46,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 9743038);