#8924670

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	#55310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831866);