#9023538

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	#58087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,e^2=g,f^2=gi,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, 8930734);