#508373

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(7,10)(8,9)(11,12)(14,19)(20,34)(21,35)(22,32)(23,41)(29,43)(37,40)(39,42);(6,9)(10,12)(13,14,16,19)(15,23,24,41)(17,29,27,43)(18,32,28,22)(20,36,34,25)(21,38,35,31)(26,42,33,39)(30,40,44,37);(2,4)(7,11)(10,12)(13,15)(14,20)(16,24)(17,30)(18,33)(19,34)(21,39)(22,40)(23,29)(25,31)(26,28)(27,44)(32,37)(35,42)(36,38)(41,43);(5,7,8,11)(6,10,9,12)(15,24)(20,34)(23,41)(25,36)(26,33)(30,44)(37,40)(39,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=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, 415569);