#8322348

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	#44912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8229544);