#8436917

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,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, 8344113);