#8139468

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	#49725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,e^2=g,f^2=h,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, 8046664);