#6432620

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339816);