#8180499

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

3993 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:

(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,20)(21,30)(22,25)(23,26)(24,27)(28,31)(29,32)(35,41)(36,44)(39,42)(40,45);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19)(18,21)(20,27)(22,28)(23,29)(24,30)(25,26)(31,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3,6,4)(2,7,10,8)(5,12,13,11)(9,15,16,14)(17,20,24,21)(18,19,27,30)(22,31,29,26)(23,25,28,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,47)(45,48);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,15,16,14)(17,21,24,20)(18,19,27,30)(22,26,29,31)(23,25,28,32);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,22,24,29)(18,25,27,32)(19,28,30,23)(20,31,21,26)(33,36,38,44)(34,40,37,45)(35,43,41,47)(39,46,42,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,23,24,28)(18,26,27,31)(19,29,30,22)(20,32,21,25)(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=h,c^2=g,d^2=g,e^2=gh,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, 8087695);