#8993642

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	#41078	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900838);