#480238

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,f^b=fg,d^c=dgi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387434);