#8661354

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

4191 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34,40,42)(35,39,41,36)(37,45,46,38)(43,44,48,47);(2,7)(3,10)(8,14)(11,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,40,41)(34,36,42,39)(37,43,46,48)(38,44,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,36,40,39)(34,35,42,41)(37,44,46,47)(38,43,45,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,37)(34,38)(35,43)(36,44)(39,47)(40,46)(41,48)(42,45);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,38)(34,37)(35,44)(36,43)(39,48)(40,45)(41,47)(42,46);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,39,40,36)(34,41,42,35)(37,44,46,47)(38,43,45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8568550);