#8393412

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

3967 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)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28)(33,34,38,39)(35,37,41,47)(36,43,46,40)(42,48,44,45);(1,3,7,4)(2,8,9,10)(5,14,16,12)(6,13,11,15)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32)(33,34,38,39)(35,37,41,47)(36,43,46,40)(42,48,44,45);(2,9)(6,11)(8,10)(13,15)(22,30)(25,29)(27,31)(28,32)(34,39)(37,47)(40,43)(45,48);(1,4,7,3)(2,10,9,8)(5,12,16,14)(6,15,11,13)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,35,38,41)(34,37,39,47)(36,44,46,42)(40,48,43,45);(1,5)(2,11)(3,12)(4,14)(6,9)(7,16)(8,13)(10,15)(17,21,23,18)(19,20,24,26)(22,29,30,25)(27,28,31,32)(33,36,38,46)(34,40,39,43)(35,42,41,44)(37,45,47,48);(1,6,7,11)(2,5,9,16)(3,13,4,15)(8,14,10,12)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,37)(34,41)(35,39)(36,45)(38,47)(40,44)(42,43)(46,48)

Polycyclic presentation:

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

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, 8300608);