#10247425

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

4174 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)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34)(35,43)(36,39)(37,41)(38,46)(40,44)(42,47)(45,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,35,39,44)(34,40,36,43)(37,47,48,38)(41,46,45,42);(2,7)(3,10)(8,15)(11,12)(34,36)(40,43)(41,45)(42,46);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36,39,34)(35,40,44,43)(37,41,48,45)(38,46,47,42);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,19)(18,24,22,26)(20,27,30,29)(25,32,28,31)(33,37,39,48)(34,41,36,45)(35,38,44,47)(40,42,43,46);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,38)(34,42)(35,37)(36,46)(39,47)(40,41)(43,45)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154621);