#9054432

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	#50125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28)(35,41)(38,42)(43,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,21)(19,26)(25,29)(27,32)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(2,6)(3,10)(4,13)(14,15)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961628);