#505208

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ei,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^c=di

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, 412404);