#8993176

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	#42144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,5)(4,13)(6,14)(7,9)(8,12)(10,11)(15,16)(17,18,23,27)(19,22,21,25)(20,26,31,24)(28,30,29,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,7,12)(2,8,9,5)(4,10,14,16)(6,11,13,15)(17,19,23,21)(18,22,27,25)(20,29,31,28)(24,30,26,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(1,4,7,14)(2,6,9,13)(3,10,12,16)(5,15,8,11)(17,20,23,31)(18,24,27,26)(19,28,21,29)(22,32,25,30)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,6)(3,10)(5,15)(7,14)(8,11)(9,13)(12,16)(17,21,23,19)(18,25,27,22)(20,28,31,29)(24,32,26,30)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5,7,8)(2,3,9,12)(4,11,14,15)(6,16,13,10)(17,20)(18,26)(19,28)(21,29)(22,30)(23,31)(24,27)(25,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,6)(2,4)(3,11)(5,16)(7,13)(8,10)(9,14)(12,15)(17,22,23,25)(18,19,27,21)(20,30,31,32)(24,29,26,28)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=i,e^2=g,f^2=gh,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, 8900372);