#418125

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325321);