#8321869

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

3849 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8229065);