#8720590

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	#55328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(34,36)(40,42)(41,44)(47,48);(3,10)(6,7)(11,16)(14,15)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(33,34,39,36)(35,40,38,42)(37,41,45,44)(43,47,46,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,36,39,34)(35,40,38,42)(37,44,45,41)(43,47,46,48);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,38,39,35)(34,42,36,40)(37,43,45,46)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=g,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, 8627786);