#8461586

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4121 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=h,e^2=hi,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, 8368782);