#8446098

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34,38,40)(35,41,43,37)(36,39,45,46)(42,47,48,44);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(18,23)(19,26)(24,30)(27,32)(33,35,38,43)(34,37,40,41)(36,42,45,48)(39,44,46,47);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,37)(36,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,41)(36,39)(37,43)(38,40)(42,47)(44,48)(45,46);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36,38,45)(34,39,40,46)(35,42,43,48)(37,44,41,47);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,37,38,41)(34,35,40,43)(36,44,45,47)(39,42,46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,e^2=hi,f^2=g,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, 8353294);