#498161

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	#33516	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

748 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)(5,6,11,16,10,13,28,17)(7,18,22,34,21,24,23,36)(8,14,26,9,25,30,15,29)(12,31,19,33,32,27,35,20);(5,7,11,22,10,21,28,23)(6,12,16,19,13,32,17,35)(8,24,26,36,25,18,15,34)(9,27,30,20,29,31,14,33);(2,4)(6,13)(8,25)(12,32)(15,26)(16,17)(18,24)(19,35)(34,36);(5,8)(6,14)(7,19)(9,16)(10,25)(11,26)(12,23)(13,30)(15,28)(17,29)(18,33)(20,24)(21,35)(22,32)(27,34)(31,36)

Polycyclic presentation:

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

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, 405357);