#8458608

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

4191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=i,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, 8365804);