#8459124

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

3967 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)(24,31)(27,32)(29,30)(33,34)(35,41)(36,43)(37,40)(38,39)(42,45)(44,48)(46,47);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(34,39)(36,41)(40,47)(44,45);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,45)(38,41)(39,43)(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)(18,24,26,31)(19,27,25,32)(21,29,23,30);(1,6,7,2)(3,10,5,8)(4,9,14,12)(11,15,13,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,37,38,46)(34,40,39,47)(35,42,43,48)(36,44,41,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8366320);