#190177

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

987 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:

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

Polycyclic presentation:

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

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, 97373);