#8688369

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

4245 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(20,28)(23,31)(27,32)(29,30)(33,34,38,42)(35,39,44,41)(36,45,37,40)(43,48,46,47);(1,3,7,6)(2,5,10,11)(4,12,15,13)(8,14,9,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,35)(34,39)(36,46)(37,43)(38,44)(40,48)(41,42)(45,47);(1,4)(2,8)(3,12)(5,14)(6,13)(7,15)(9,10)(11,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,36,38,37)(34,40,42,45)(35,43,44,46)(39,47,41,48);(1,4)(2,9)(3,13)(5,14)(6,12)(7,15)(8,10)(11,16)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,34)(35,41)(36,40)(37,45)(38,42)(39,44)(43,48)(46,47);(1,5,7,11)(2,3,10,6)(4,14,15,16)(8,12,9,13)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,35)(34,41)(36,43)(37,46)(38,44)(39,42)(40,48)(45,47);(1,6,7,3)(2,5,10,11)(4,13,15,12)(8,14,9,16)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,37,38,36)(34,40,42,45)(35,43,44,46)(39,48,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595565);