#8322020

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,34,38,37)(35,39,43,41)(36,44,46,40)(42,48,45,47);(2,7)(3,12)(9,14)(10,13)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,34,38,37)(35,41,43,39)(36,40,46,44)(42,48,45,47);(2,7)(5,16)(8,11)(9,14)(21,29)(24,31)(27,28)(30,32)(34,37)(39,41)(40,44)(47,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,37,38,34)(35,41,43,39)(36,44,46,40)(42,48,45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8229216);