#8898642

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	#47589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805838);