#8425357

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

4225 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)(18,23)(19,28)(20,27)(30,31)(34,39)(40,41)(42,45)(47,48);(3,10)(7,8)(11,16)(12,15)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,25)(26,31)(27,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,19)(18,25)(20,29)(21,31)(22,28)(23,24)(26,30)(27,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,20,22,27)(18,26,23,21)(19,29,28,32)(24,31,25,30)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,21,22,26)(18,27,23,20)(19,30,28,31)(24,32,25,29)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=i,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, 8332553);