#8943763

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850959);