#8459106

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

4184 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(20,28)(24,31)(27,32)(29,30)(33,34,40,38)(35,44,47,41)(36,48,42,37)(39,43,46,45);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,41)(36,45)(37,39)(38,44)(40,47)(42,43)(46,48);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36)(34,37)(35,45)(38,48)(39,41)(40,42)(43,47)(44,46);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37,40,48)(34,42,38,36)(35,46,47,39)(41,45,44,43);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,38,40,34)(35,44,47,41)(36,48,42,37)(39,45,46,43);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,39,40,46)(34,43,38,45)(35,37,47,48)(36,41,42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=hi,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, 8366302);