#8475859

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	#36776	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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)(3,9)(4,8)(5,16)(6,10)(7,11)(12,15)(13,14)(20,28)(24,29)(27,32)(30,31)(33,34,38,39)(35,36,41,40)(37,42,46,45)(43,44,48,47);(1,3,7,4)(2,8,11,9)(5,14,10,12)(6,13,16,15)(17,18,22,26)(19,21,25,23)(20,29,28,24)(27,31,32,30)(34,39)(36,40)(42,45)(44,47);(1,3,7,4)(2,9,11,8)(5,12,10,14)(6,13,16,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4,7,3)(2,9,11,8)(5,14,10,12)(6,13,16,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(1,5,7,10)(2,6,11,16)(3,12,4,14)(8,15,9,13)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,6)(2,10)(3,13)(4,15)(5,11)(7,16)(8,12)(9,14)(17,21)(18,25)(19,26)(20,30)(22,23)(24,32)(27,29)(28,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

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

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, 8383055);