#10320390

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10227586);