#8680009

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	#52294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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)(10,16)(11,13)(18,24)(19,20)(21,29)(30,31);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,13)(15,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,30)(28,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,44)(45,47)(46,48);(17,19,23,20)(18,25,24,22)(21,27,29,28)(26,30,32,31)(33,35)(34,37)(36,43)(38,41)(39,46)(40,42)(44,47)(45,48);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,30,32,31)(33,36,38,45)(34,39,42,47)(35,43,41,48)(37,46,40,44);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,21,23,29)(18,26,24,32)(19,27,20,28)(22,31,25,30)(33,35)(34,40)(36,43)(37,42)(38,41)(39,44)(45,48)(46,47);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,22,23,25)(18,20,24,19)(21,30,29,31)(26,27,32,28)(33,37,38,40)(34,41,42,35)(36,44,45,46)(39,43,47,48)

Polycyclic presentation:

2^9|c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587205);