#8095870

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	#36530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,18,23,26)(19,21,24,20)(22,25,30,32)(27,29,31,28)(33,34,40,44)(35,39,41,36)(37,45,48,42)(38,43,46,47);(4,12)(7,14)(10,16)(13,15)(17,19,23,24)(18,20,26,21)(22,27,30,31)(25,28,32,29)(33,35,40,41)(34,36,44,39)(37,46,48,38)(42,47,45,43);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,21)(18,19,26,24)(22,28,30,29)(25,27,32,31)(33,36,40,39)(34,41,44,35)(37,43,48,47)(38,42,46,45);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,37)(34,42)(35,38)(36,43)(39,47)(40,48)(41,46)(44,45);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,23,20)(18,24,26,19)(22,28,30,29)(25,27,32,31)(33,38)(34,43)(35,37)(36,42)(39,45)(40,46)(41,48)(44,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39,40,36)(34,41,44,35)(37,47,48,43)(38,42,46,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003066);