#8634847

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	#45508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34,40,38)(35,39,43,41)(36,42,37,47)(44,46,45,48);(1,3,7,10)(2,8,5,6)(4,12,15,11)(9,16,13,14)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(33,35,40,43)(34,41,38,39)(36,44,37,45)(42,48,47,46);(2,5)(3,10)(9,13)(11,12)(19,26)(21,24)(27,32)(28,31)(33,36,40,37)(34,42,38,47)(35,44,43,45)(39,46,41,48);(1,4,7,15)(2,9,5,13)(3,11,10,12)(6,16,8,14)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,37,40,36)(34,42,38,47)(35,45,43,44)(39,46,41,48);(1,5,7,2)(3,8,10,6)(4,13,15,9)(11,14,12,16)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,38,40,34)(35,41,43,39)(36,42,37,47)(44,46,45,48);(1,6,7,8)(2,10,5,3)(4,14,15,16)(9,11,13,12)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,39,40,41)(34,43,38,35)(36,46,37,48)(42,45,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8542043);