#8425367

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

4210 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)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34)(35,38)(36,47)(37,39)(40,45)(41,44)(42,46)(43,48);(3,10)(7,8)(11,16)(12,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,40,44)(34,41,45,38)(36,39,46,43)(37,47,48,42);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36)(34,42)(35,39)(37,41)(38,48)(40,46)(43,44)(45,47);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,37,40,48)(34,43,45,39)(35,42,44,47)(36,41,46,38);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,38,40,41)(34,44,45,35)(36,48,46,37)(39,42,43,47);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,39,40,43)(34,37,45,48)(35,46,44,36)(38,42,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,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, 8332563);