#189041

order	512
nilpotency class	3
isoclinism	isoclinic class of #188968
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,f^c=fh,g^c=ghi

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, 96237);