#8534886

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

4178 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8442082);