#8757275

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	#45805	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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,20)(19,25)(21,29)(30,32)(33,34,39,42)(35,41,37,40)(36,45,47,38)(43,44,46,48);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,35,39,37)(34,40,42,41)(36,46,47,43)(38,48,45,44);(4,13)(8,12)(10,16)(14,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,37,39,35)(34,41,42,40)(36,46,47,43)(38,48,45,44);(1,5,6,2)(3,7,11,9)(4,12,13,8)(10,14,16,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47);(1,3)(2,9)(4,10)(5,7)(6,11)(8,15)(12,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,44)(45,47)(46,48)

Polycyclic presentation:

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

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, 8664471);