#8202034

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(3,4)(5,13)(6,8)(7,15)(9,11)(10,16)(12,14)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,6,11)(2,4,8,9)(5,14,15,10)(7,16,13,12)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29)(35,43)(36,46)(39,48)(40,45);(1,2,6,8)(3,9,11,4)(5,7,15,13)(10,16,14,12)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,14)(15,16)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28);(1,5)(2,7)(3,10)(4,12)(6,15)(8,13)(9,16)(11,14)(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

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

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, 8109230);