#8479491

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	#43080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(3,7)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(18,24)(19,20)(25,29)(26,28)(33,34)(35,42)(36,41)(37,46)(38,39)(40,45)(43,44)(47,48);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,18,23,24)(19,22,20,27)(21,29,31,25)(26,32,28,30)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,47)(46,48);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,19,23,20)(18,22,24,27)(21,26,31,28)(25,30,29,32)(33,36,39,46)(34,41,38,37)(35,43,45,48)(40,47,42,44);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,20,23,19)(18,22,24,27)(21,28,31,26)(25,30,29,32)(33,37)(34,36)(35,44)(38,46)(39,41)(40,43)(42,48)(45,47);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,31)(18,25,24,29)(19,26,20,28)(22,30,27,32)(33,38,39,34)(35,40,45,42)(36,37,46,41)(43,47,48,44);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,22)(18,20)(19,24)(21,30)(23,27)(25,28)(26,29)(31,32)(33,35)(34,42)(36,43)(37,47)(38,40)(39,45)(41,44)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386687);