#453828

order	512
nilpotency class	3
isoclinism	isoclinic class of #453828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16135
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

665 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)(3,8,14,26)(4,13,19,20)(5,16)(6,12)(7,11)(9,18,27,31)(10,22)(15,25,23,17)(21,30)(24,28)(29,32);(1,3)(2,8,11,26)(4,16)(5,20,22,13)(6,23)(7,14)(9,21,27,32)(10,19)(12,17,28,25)(15,24)(18,29)(30,31);(1,4)(2,9,11,27)(3,10,14,16)(5,17)(6,18)(7,19)(8,21)(12,13,28,20)(15,29,23,30)(22,25)(24,31)(26,32);(1,5,6,21,7,22,24,32)(2,10,28,30,11,16,12,29)(3,13,23,9,14,20,15,27)(4,17,18,8,19,25,31,26)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,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, 361024);