#9851271

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

4102 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,26)(19,24,30,21)(20,22,32,25)(27,31,28,29)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,47,46,43);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,30)(18,24,26,21)(20,28,32,27)(22,29,25,31)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,4)(3,10)(9,12)(11,16)(17,20,23,32)(18,25,26,22)(19,27,30,28)(21,31,24,29)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,22)(19,28)(21,31)(23,32)(24,29)(25,26)(27,30)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,19)(20,31)(22,28)(23,24)(25,27)(26,30)(29,32)(33,38,40,36)(34,42,44,37)(35,47,48,43)(39,45,46,41);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22)(18,20)(19,29)(21,27)(23,25)(24,28)(26,32)(30,31)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=hi,f^2=g,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, 9758467);