#8445940

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	#53655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353136);