#6449803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,27,30,24)(20,25,29,32)(21,26,28,31);(2,5)(6,8)(17,19,22,30)(18,24,23,27)(20,31,29,26)(21,25,28,32);(10,13)(14,16)(17,20,22,29)(18,25,23,32)(19,28,30,21)(24,26,27,31);(9,11,12,15)(10,14,13,16)(17,21,22,28)(18,26,23,31)(19,29,30,20)(24,25,27,32);(1,3,4,7)(2,6,5,8)(19,30)(20,29)(24,27)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6356999);