#8731118

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

3912 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(2,7)(3,9)(8,11)(10,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,7)(3,9)(8,11)(10,16)(19,27)(23,24)(28,32)(29,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,20)(18,21)(19,28)(22,25)(23,29)(24,30)(26,31)(27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8638314);