#190431

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2283 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

959 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,g^b=gh,d^c=di,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, 97627);