#451178

order	512
nilpotency class	3
isoclinism	isoclinic class of #451178
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15987
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

785 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)(3,8,14,24)(4,16)(5,12)(6,10)(7,11)(9,18)(13,29,17,32)(15,27,20,23)(19,28)(21,26)(22,31,25,30);(1,3,5,15,7,14,6,20)(2,8,10,23,11,24,12,27)(4,17,18,31,19,13,21,30)(9,25,16,32,26,22,28,29);(1,4)(2,9,11,26)(3,13,14,17)(5,18)(6,21)(7,19)(8,22)(10,16,12,28)(15,30,20,31)(23,29)(24,25)(27,32);(2,10)(3,14)(5,6)(8,23)(9,16)(11,12)(13,17)(18,21)(22,29)(24,27)(25,32)(26,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358374);