#8436813

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=h,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, 8344009);