#8188201

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

4056 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)(3,10)(4,8)(5,6)(7,9)(11,16)(12,15)(13,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,39)(36,41)(37,45)(38,40)(42,46)(43,44)(47,48);(1,3,7,4)(2,8,9,10)(5,12,16,14)(6,13,11,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,41)(34,39,40,36)(37,43,46,48)(42,47,45,44);(2,9)(6,11)(8,10)(13,15)(22,30)(25,28)(27,29)(31,32)(34,40)(35,41)(42,45)(43,48);(1,4,7,3)(2,10,9,8)(5,14,16,12)(6,15,11,13)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,38,39)(34,41,40,35)(37,44,46,47)(42,48,45,43);(1,5)(2,11)(3,12)(4,14)(6,9)(7,16)(8,15)(10,13)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,38,46)(34,42,40,45)(35,43,41,48)(36,44,39,47);(1,6,7,11)(2,5,9,16)(3,13,4,15)(8,12,10,14)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|b^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095397);