#8534917

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	#55218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=hi,f^2=g,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, 8442113);