#9029629

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,e^2=h,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, 8936825);