#8127345

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	#48669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034541);