#8897950

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	#38672	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=h,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, 8805146);