#9481535

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	#53934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(3,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,32)(27,31)(28,29)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(2,8)(5,12)(9,10)(13,15)(17,19,23,21)(18,24,26,20)(22,27,32,29)(25,31,30,28)(34,35)(40,43)(41,42)(47,48);(1,3,7,4)(2,9,8,10)(5,13,12,15)(6,16,11,14)(17,20,23,24)(18,21,26,19)(22,31,32,28)(25,27,30,29)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,7,3)(2,10,8,9)(5,13,12,15)(6,16,11,14)(17,21,23,19)(18,24,26,20)(22,29,32,27)(25,31,30,28)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,5,7,12)(2,11,8,6)(3,13,4,15)(9,14,10,16)(17,18)(19,24)(20,21)(22,25)(23,26)(27,31)(28,29)(30,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,6,7,11)(2,12,8,5)(3,14,4,16)(9,13,10,15)(17,22,23,32)(18,25,26,30)(19,27,21,29)(20,28,24,31)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388731);