#8436782

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	#41311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(18,24)(19,20)(26,31)(27,28)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,29)(30,31)(35,44)(36,46)(39,43)(40,47);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19,23,20)(18,25,24,22)(21,27,30,28)(26,32,31,29)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,47,48);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,20,23,19)(18,25,24,22)(21,28,30,27)(26,32,31,29)(33,36)(34,40)(35,45)(37,44)(38,46)(39,41)(42,47)(43,48);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,21,23,30)(18,26,24,31)(19,27,20,28)(22,29,25,32)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,22,23,25)(18,20,24,19)(21,29,30,32)(26,28,31,27)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=g,e^2=h,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, 8343978);