#363335

order	512
nilpotency class	3
isoclinism	isoclinic class of #360200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2311
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13938
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(4,16,18,8)(5,9)(6,11)(7,14)(10,19)(13,26,28,32)(15,21)(17,25,30,24)(20,31,23,27)(22,29);(1,3,5,14,6,15,19,29)(2,7,10,12,11,22,9,21)(4,13,17,27,18,28,30,31)(8,23,25,32,16,20,24,26);(1,4)(2,8)(3,13)(5,17)(6,18)(7,20)(9,24)(10,25)(11,16)(12,26)(14,27)(15,28)(19,30)(21,32)(22,23)(29,31);(2,9,11,10)(7,21,22,12)(8,24,16,25)(20,32,23,26)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 270531);