#8458821

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	#39910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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, 8366017);