#8095779

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

4115 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:

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

Polycyclic presentation:

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

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, 8002975);