#8437741

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,30,32)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,46)(47,48);(1,3,8,15)(2,9,11,12)(4,13,5,6)(7,14,16,10)(17,19)(18,25)(20,30)(21,23)(22,29)(24,31)(26,32)(27,28)(35,46)(36,47)(40,45)(41,48);(1,4,8,5)(2,7,11,16)(3,13,15,6)(9,14,12,10)(17,20,24,21)(18,26,22,27)(19,30,31,23)(25,32,29,28)(33,35,39,46)(34,40,44,45)(36,43,47,37)(38,41,42,48);(1,5,8,4)(2,7,11,16)(3,6,15,13)(9,14,12,10)(17,21,24,20)(18,27,22,26)(19,30,31,23)(25,32,29,28)(33,36)(34,41)(35,43)(37,46)(38,45)(39,47)(40,42)(44,48);(1,6)(2,10)(3,4)(5,15)(7,9)(8,13)(11,14)(12,16)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,32,30,28)(33,37,39,43)(34,42,44,38)(35,36,46,47)(40,48,45,41);(1,7)(2,5)(3,14)(4,11)(6,9)(8,16)(10,15)(12,13)(17,23,24,30)(18,28,22,32)(19,21,31,20)(25,27,29,26)(33,38,39,42)(34,43,44,37)(35,41,46,48)(36,40,47,45)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=h,f^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, 8344937);