#8459112

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

4190 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(20,28)(24,31)(27,32)(29,30)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,46,48);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,42)(34,41,44,37)(35,46,47,38)(40,48,45,43);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,34)(35,45)(36,41)(37,42)(38,48)(39,44)(40,47)(43,46);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,41)(34,42,44,36)(35,43,47,48)(38,45,46,40);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,46)(34,43,44,48)(35,42,47,36)(37,45,41,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,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, 8366308);