#8460721

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	#40356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8367917);