#8436718

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343914);