#10412070

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

4347 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)(3,4,11,7)(5,8,14,16)(10,12,15,13)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,3,6,11)(2,7,9,4)(5,10,14,15)(8,13,16,12)(18,23)(20,25)(24,29)(28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,43)(42,47)(44,45)(46,48);(1,4)(2,3)(5,13)(6,7)(8,15)(9,11)(10,16)(12,14)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,31)(28,29)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,14)(2,8,9,16)(3,10,11,15)(4,12,7,13)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319266);