#8425358

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

4205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332554);