#8993555

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	#35470	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=ghi,f^2=gh,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, 8900751);