#8505348

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=gi,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, 8412544);