#404123

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311319);