#505697

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

547 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)(5,6,8,10)(7,9,11,12)(13,14,18,23)(15,25,27,42)(16,29,31,43)(17,33,30,44)(19,35,34,24)(20,36,37,28)(21,40,39,32)(22,41,38,26);(5,7)(6,9)(8,11)(10,12)(13,15,26,32,19,16,30,36,18,27,41,40,34,31,17,28)(14,20,33,43,24,39,38,42,23,37,44,29,35,21,22,25);(2,4)(7,11)(9,12)(15,27)(16,31)(20,37)(21,39)(25,42)(28,36)(29,43)(32,40);(13,16,26,36,19,27,30,40,18,31,41,28,34,15,17,32)(14,21,33,25,24,20,38,43,23,39,44,42,35,37,22,29)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,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, 412893);