#498346

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	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33604	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,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, 405542);