#8445718

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

4062 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)(3,10,14,8)(4,6,5,15)(11,16,12,13)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,38,42)(35,43,44,39)(36,45,47,40)(37,41,46,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(19,29)(20,31)(24,28)(25,30)(33,35,38,44)(34,39,42,43)(36,46,47,37)(40,48,45,41);(1,4,7,5)(2,6,9,15)(3,11,14,12)(8,13,10,16)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,47)(46,48);(1,5,7,4)(2,6,9,15)(3,12,14,11)(8,13,10,16)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36,38,47)(34,40,42,45)(35,37,44,46)(39,41,43,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,6)(2,5)(3,13)(4,9)(7,15)(8,12)(10,11)(14,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37)(34,41)(35,36)(38,46)(39,40)(42,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=i,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, 8352914);