#201208

order	512
nilpotency class	3
isoclinism	isoclinic class of #201085
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108404);