#8843436

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	#41295	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

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

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, 8750632);