#8425348

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332544);