#9794284

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	#53832	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9701480);