#8139495

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

3901 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046691);