#6447644

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3101 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)(7,11,12,10)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,37)(26,34)(28,38)(29,35)(30,36)(31,42)(33,41)(39,43)(40,44);(1,2)(3,4)(6,9)(10,11)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,38)(16,25,31,41,30,40,29,39)(21,33,35,37,36,43,42,44);(2,4)(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,35)(23,36)(24,37)(26,39)(27,41)(28,40)(32,42)(34,43)(38,44);(2,4)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,30,31)(20,24,34,38)(21,35,36,42)(25,39,40,41)(33,37,43,44);(16,30)(21,36)(25,40)(29,31)(33,43)(35,42)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6354840);