#8461616

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

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8368812);