#8436780

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	#41157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(18,24)(19,20)(26,30)(27,29)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(35,44)(36,46)(39,43)(40,47);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,47,48);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,36)(34,40)(35,45)(37,44)(38,46)(39,41)(42,47)(43,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|b^2=hi,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, 8343976);