#189381

order	512
nilpotency class	3
isoclinism	isoclinic class of #189323
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,f^c=fhi,g^c=gh

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, 96577);