#8180396

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

4135 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087592);