#9039270

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	#56189	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946466);