#8180463

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,10)(4,7)(5,12)(6,9)(8,14)(11,13)(15,16)(17,18,23,22)(19,24,29,28)(20,30,25,21)(26,32,31,27)(35,41)(36,44)(39,42)(40,45);(2,4)(3,9)(8,12)(11,15)(17,19,23,29)(18,24,22,28)(20,26,25,31)(21,27,30,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20)(18,21)(19,26)(22,30)(23,25)(24,27)(28,32)(29,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,47)(45,48);(1,4,7,2)(3,10,9,6)(5,12,14,8)(11,16,15,13)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32);(1,5,7,14)(2,8,4,12)(3,11,9,15)(6,13,10,16)(17,21,23,30)(18,25,22,20)(19,27,29,32)(24,31,28,26)(33,36,38,44)(34,40,37,45)(35,43,41,47)(39,46,42,48);(1,6,7,10)(2,9,4,3)(5,13,14,16)(8,15,12,11)(17,22,23,18)(19,28,29,24)(20,30,25,21)(26,32,31,27)(33,37,38,34)(35,39,41,42)(36,45,44,40)(43,46,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=h,f^2=h,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, 8087659);