#8193896

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

3998 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,14)(15,16)(33,34,38,37)(35,39,43,42)(36,44,45,40)(41,47,48,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,37)(35,39,43,42)(36,40,45,44)(41,46,48,47);(2,7)(6,9)(8,12)(14,15)(18,22)(23,25)(24,29)(30,31)(34,37)(39,42)(40,44)(46,47);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,38,43)(34,39,37,42)(36,41,45,48)(40,46,44,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,38,45)(34,40,37,44)(35,41,43,48)(39,46,42,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,38,34)(35,42,43,39)(36,44,45,40)(41,47,48,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gi,e^2=i,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, 8101092);