#8813996

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	#51815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,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, 8721192);