#8963674

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	#54964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870870);