#8898544

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	#36921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,7,11,16)(4,12,5,8)(10,15,13,14)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,39,44)(35,43,46,40)(36,47,48,41)(37,38,45,42);(1,3,6,11)(2,7,9,16)(4,13,5,10)(8,15,12,14)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,35,39,46)(34,40,44,43)(36,45,48,37)(38,41,42,47);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,14,16,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,36,39,48)(34,41,44,47)(35,45,46,37)(38,40,42,43);(1,5,6,4)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,42)(35,36)(38,44)(39,45)(40,47)(41,43)(46,48);(1,3)(2,7)(4,10)(5,13)(6,11)(8,14)(9,16)(12,15)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,35)(34,43)(36,45)(37,48)(38,41)(39,46)(40,44)(42,47);(1,2)(3,7)(4,12)(5,8)(6,9)(10,15)(11,16)(13,14)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38,39,42)(34,37,44,45)(35,41,46,47)(36,43,48,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gh,f^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, 8805740);