#188168

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
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×#562	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2426 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,24,20,25)(19,27,28,32,29,23,30,31);(4,12)(7,14)(10,16)(13,15)(17,19,22,29)(18,23,24,27)(20,28,21,30)(25,31,26,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,24)(20,21)(23,27)(28,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95364);