#8436720

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=h,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, 8343916);