#8112548

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8019744);