#10407834

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

4314 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10315030);