#8137114

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044310);