#6432626

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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2794 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fg,e^2=i,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, 6339822);