#8125449

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	#53102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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)(3,9)(4,6)(5,12)(7,11)(8,13)(10,14)(15,16)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(33,34)(35,41)(36,46)(37,38)(39,48)(40,42)(43,45)(44,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,35,40,39)(34,41,42,48)(36,44,37,43)(38,45,46,47);(2,4)(7,9)(8,12)(14,15)(19,26)(21,24)(27,32)(29,31)(33,36,40,37)(34,38,42,46)(35,43,39,44)(41,47,48,45);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37,40,36)(34,38,42,46)(35,44,39,43)(41,47,48,45);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,38)(34,36)(35,45)(37,42)(39,47)(40,46)(41,44)(43,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,39,40,35)(34,41,42,48)(36,44,37,43)(38,47,46,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8032645);