#8650382

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8557578);