#500434

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 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)(7,11)(8,9)(10,12)(13,14,18,22)(15,19,25,33)(16,20,29,37)(17,21,28,36)(23,31,40,43)(24,32,30,38)(26,34,39,42)(27,35,41,44);(6,9)(10,11)(15,23)(16,26)(17,28)(19,31)(20,34)(21,36)(24,41)(25,40)(27,30)(29,39)(32,44)(33,43)(35,38)(37,42);(2,4)(13,15,18,25)(14,19,22,33)(16,27,29,41)(17,30,28,24)(20,35,37,44)(21,38,36,32)(23,39,40,26)(31,42,43,34);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 407630);