#9054415

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3777 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)(3,7)(4,11)(5,13)(6,8)(9,14)(10,16)(12,15)(18,23)(21,24)(25,30)(29,31)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,3,6,4)(2,7,8,11)(5,10,14,12)(9,15,13,16)(20,28)(25,30)(27,32)(29,31)(34,37)(35,41)(40,44)(43,47);(2,8)(3,4)(5,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,4,6,3)(2,7,8,11)(5,12,14,10)(9,15,13,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,5,6,14)(2,9,8,13)(3,10,4,12)(7,15,11,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961611);