#8993152

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	#55473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8900348);