#6439719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei

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, 6346915);