#361382

order	512
nilpotency class	3
isoclinism	isoclinic class of #361285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2290
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13896
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1065 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)(3,13,16,24)(4,8,15,9)(5,19,21,32)(6,11)(7,12)(10,20,26,30)(14,25)(17,22)(18,27)(23,29)(28,31);(1,3)(2,8,12,9)(4,17)(5,14)(6,15)(7,16)(10,23,26,25)(11,13,22,24)(18,30)(19,27,32,31)(20,28)(21,29);(1,4,6,3,7,15,17,16)(2,9,11,24,12,8,22,13)(5,18,20,14,21,28,30,29)(10,25,19,31,26,23,32,27);(1,5)(2,10,12,26)(3,14)(4,18)(6,20)(7,21)(8,23,9,25)(11,19,22,32)(13,27,24,31)(15,28)(16,29)(17,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268578);