#8928619

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	#36811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,31)(30,32)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,24)(20,26)(21,28)(22,27)(23,29)(25,32)(30,31)(37,45)(41,44)(43,46)(47,48);(3,11)(4,12)(6,16)(8,13)(17,20,23,31)(18,25,27,28)(19,26,29,30)(21,24,32,22)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,21,23,32)(18,26,27,30)(19,28,29,25)(20,22,31,24)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,22)(18,19)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32)(33,37,39,45)(34,41,36,44)(35,43,38,46)(40,47,42,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,19)(18,22)(20,30)(21,28)(23,29)(24,27)(25,32)(26,31)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835815);