#8720324

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 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)(3,6)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34,39,42)(35,41,37,40)(36,45,47,38)(43,44,46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(33,35,39,37)(34,40,42,41)(36,43,47,46)(38,44,45,48);(1,2,5,9)(3,7,11,6)(4,8,14,12)(10,16,13,15)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,35,39,37)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,2)(3,7)(4,12)(5,9)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,36,39,47)(34,38,42,45)(35,43,37,46)(40,44,41,48);(1,3)(2,7)(4,13)(5,11)(6,9)(8,15)(10,14)(12,16)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,37,39,35)(34,40,42,41)(36,46,47,43)(38,44,45,48);(1,4,5,14)(2,8,9,12)(3,10,11,13)(6,15,7,16)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627520);