#8688279

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

4247 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,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, 8595475);