#8174410

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081606);