#8893598

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	#36926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800794);