#8438167

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(20,28)(24,31)(26,32)(29,30)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,38)(34,40,44,43)(36,47,45,37)(41,42,46,48);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36)(34,41)(35,37)(38,47)(39,45)(40,48)(42,43)(44,46);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,47)(34,42,44,48)(35,45,38,36)(40,41,43,46);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,20,21,28)(18,24,25,31)(19,26,27,32)(22,29,23,30)(33,38,39,35)(34,43,44,40)(36,47,45,37)(41,42,46,48);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,34)(35,40)(36,41)(37,48)(38,43)(39,44)(42,47)(45,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345363);