#502894

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

859 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410090);