#8436687

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(26,30)(27,29)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,45)(46,48);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(33,35,39,46)(34,40,43,48)(36,37,42,44)(38,45,47,41);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,36,39,42)(34,38,43,47)(35,44,46,37)(40,41,48,45);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,48)(45,47);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28)(33,37,39,44)(34,41,43,45)(35,42,46,36)(38,48,47,40);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,38)(34,42)(35,45)(36,43)(37,48)(39,47)(40,44)(41,46)

Polycyclic presentation:

2^9|b^2=gi,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, 8343883);