#8932732

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

4133 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

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, 8839928);