#190328

order	512
nilpotency class	3
isoclinism	isoclinic class of #190218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
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

897 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)(3,8,4,15)(5,10)(6,12)(7,11)(9,13,16,14)(17,18,21,24,22,25,20,30)(19,26,28,23,29,31,27,32);(1,3,6,4)(2,8,12,15)(5,13,7,14)(9,10,16,11);(1,4,6,3)(2,9)(5,14,7,13)(8,10)(11,15)(12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30);(1,5,6,7)(2,10,12,11)(3,13,4,14)(8,16,15,9)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,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, 97524);