#9847624

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	#48893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=h,e^2=ghi,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, 9754820);