#188158

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
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

2338 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:

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

Polycyclic presentation:

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

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, 95354);