#8436788

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	#51519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(34,35)(36,43)(40,42)(44,45);(2,7)(3,10)(8,14)(11,16)(19,28)(20,30)(23,27)(24,31)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,22,28)(18,23,26,27)(20,29,30,21)(24,25,31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20)(18,24)(19,29)(21,28)(22,30)(23,25)(26,31)(27,32)(33,36)(34,39)(35,41)(37,45)(38,43)(40,48)(42,47)(44,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,22,29)(18,25,26,32)(19,20,28,30)(23,31,27,24);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,18)(19,23)(20,31)(21,25)(22,26)(24,30)(27,28)(29,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,e^2=gh,f^2=g,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, 8343984);