#6445386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3036 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6352582);