#8747887

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655083);