#9481450

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	#53228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3776 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,27)(20,31)(21,25)(22,23)(24,26)(28,30)(29,32)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(2,7)(4,8)(10,14)(12,15)(17,19,24,21)(18,25,26,27)(20,28,22,29)(23,30,31,32)(34,39)(35,42)(41,44)(43,48);(5,13)(10,14)(11,16)(12,15)(17,20,24,22)(18,23,26,31)(19,28,21,29)(25,30,27,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,21,24,19)(18,25,26,27)(20,28,22,29)(23,32,31,30)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,22,24,20)(18,23,26,31)(19,29,21,28)(25,30,27,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,23)(18,20)(19,30)(21,32)(22,26)(24,31)(25,28)(27,29)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^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, 9388646);