#9668322

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28)(34,38)(36,44)(39,41)(42,48);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,9)(4,11)(8,16)(12,14)(17,19,22,24)(18,23,26,20)(21,27,31,30)(25,32,29,28)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,19)(18,20)(21,30)(22,24)(23,26)(25,32)(27,31)(28,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,21,22,31)(18,25,26,29)(19,27,24,30)(20,28,23,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575518);