#8425662

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332858);