#361049

order	512
nilpotency class	3
isoclinism	isoclinic class of #360226
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2352
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13887
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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)(4,17,20,9)(5,10)(6,11)(7,12)(8,15)(14,29,31,26)(16,24)(18,23,30,32)(19,28,22,27)(21,25);(1,3,5,15,7,16,6,21)(2,8,11,13,12,25,10,24)(4,18,19,14,20,30,22,31)(9,26,28,32,17,29,27,23);(1,4)(2,9)(3,14)(5,19)(6,22)(7,20)(8,23)(10,27)(11,28)(12,17)(13,26)(15,30)(16,31)(18,21)(24,29)(25,32);(2,10,12,11)(8,24,25,13)(9,27,17,28)(23,29,32,26)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=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, 268245);