#188097

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,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, 95293);