#8137125

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	#58245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=gh,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, 8044321);