#8806892

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714088);