#453918

order	512
nilpotency class	3
isoclinism	isoclinic class of #453917
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16222
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

667 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)(3,12)(4,6)(5,9,20,24)(7,11)(8,15)(10,18)(13,16)(14,23,30,26)(17,27)(19,21,25,22)(28,31,32,29);(1,3)(2,8,11,12)(4,17,10,13)(5,14,20,30)(6,16)(7,15)(9,23)(18,27)(19,32)(21,29,22,31)(24,26)(25,28);(1,4,18,11,7,10,6,2)(3,13,27,8,15,17,16,12)(5,19,21,24,20,25,22,9)(14,28,29,23,30,32,31,26);(1,5,6,22,7,20,18,21)(2,9,10,25,11,24,4,19)(3,14,27,29,15,30,16,31)(8,23,17,32,12,26,13,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=efgi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 361114);