#436188

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 343384);