#498360

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405556);