#9018151

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	#48063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8925347);