#8534885

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	#38283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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)(3,12)(4,6)(5,15)(7,10)(8,11)(9,14)(13,16)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,9)(4,13)(5,10)(6,7)(8,14)(11,12)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,4,8,5)(2,6,11,15)(3,13,14,10)(7,12,16,9)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5,8,4)(2,6,11,15)(3,10,14,13)(7,12,16,9)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,6)(2,4)(3,7)(5,11)(8,15)(9,13)(10,12)(14,16)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,7)(2,10)(3,6)(4,12)(5,9)(8,16)(11,13)(14,15)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442081);