#426985

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15362
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 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)(3,13,17,29)(4,9)(5,20)(6,10)(7,11)(8,16,28,18)(12,25)(14,22,19,31)(15,27,26,32)(21,23)(24,30);(1,3,6,18,7,17,25,16)(2,8,10,29,11,28,12,13)(4,19,20,27,21,14,30,32)(5,22,23,15,24,31,9,26);(1,4)(2,9)(3,14,17,19)(5,10)(6,20)(7,21)(8,26,28,15)(11,23)(12,24)(13,31,29,22)(16,27,18,32)(25,30);(1,5,6,23,7,24,25,9)(2,4,10,20,11,21,12,30)(3,15,16,22,17,26,18,31)(8,27,13,19,28,32,29,14)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334181);