#8963624

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	#50117	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=i,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, 8870820);