#9003921

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	#40733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=gi,f^2=gh,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, 8911117);