#8184907

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	#40007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(3,6)(4,8)(5,13)(7,10)(9,15)(11,16)(12,14)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3,7,4)(2,8,10,6)(5,11,15,12)(9,16,13,14)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,4,7,3)(2,8,10,6)(5,12,15,11)(9,16,13,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5,7,15)(2,9,10,13)(3,11,4,12)(6,14,8,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,6,7,8)(2,3,10,4)(5,14,15,16)(9,11,13,12)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092103);