#8813993

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	#38534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721189);