#382356

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14332
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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)(3,11,15,27)(4,8,18,24)(5,9)(6,10)(7,14,20,28)(12,25)(13,26)(16,22,17,23)(19,30)(21,29)(31,32);(1,3,12,28,6,15,5,14)(2,7,9,11,10,20,25,27)(4,13,16,29,18,31,17,30)(8,21,23,26,24,19,22,32);(1,4)(2,8)(3,13)(5,16)(6,18)(7,19)(9,22)(10,24)(11,26)(12,17)(14,29)(15,31)(20,21)(23,25)(27,32)(28,30);(4,16,18,17)(8,22,24,23)(13,29,31,30)(19,26,21,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=fgh,e^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=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, 289552);