#496612

order	512
nilpotency class	4
isoclinism	isoclinic class of #496611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21888
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60714	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ehi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403808);