#8159809

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=gh,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, 8067005);