#8688275

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595471);