#188794

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,f^c=fh,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95990);