#9029758

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,e^2=h,f^2=gi,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, 8936954);