#9782990

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	#56648	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18,23,26)(19,24,30,21)(20,22,32,25)(27,31,28,29)(33,34,39,37)(35,43,44,40)(36,45,46,41)(38,42,48,47);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19,23,30)(18,24,26,21)(20,28,32,27)(22,29,25,31)(33,35,39,44)(34,40,37,43)(36,42,46,47)(38,41,48,45);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,20,23,32)(18,25,26,22)(19,27,30,28)(21,31,24,29)(33,34,39,37)(35,40,44,43)(36,41,46,45)(38,42,48,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,22)(19,28)(21,31)(23,32)(24,29)(25,26)(27,30)(33,36,39,46)(34,41,37,45)(35,42,44,47)(38,40,48,43);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,31)(22,28)(23,24)(25,27)(26,30)(29,32)(33,37,39,34)(35,40,44,43)(36,45,46,41)(38,42,48,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,22)(18,20)(19,29)(21,27)(23,25)(24,28)(26,32)(30,31)(33,38,39,48)(34,42,37,47)(35,41,44,45)(36,40,46,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9690186);