#8746761

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	#42565	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,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, 8653957);