#182126

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1143 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,25,22,20,24,29)(19,23,28,31,27,26,30,32);(1,3,5,13,6,12,4,11)(2,7,8,15,9,10,14,16)(17,19)(18,23)(20,26)(21,28)(22,27)(24,30)(25,31)(29,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20,21,29,22,18,24,25)(19,26,28,32,27,23,30,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(19,27)(23,26)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=h,b^a=be,g^a=gh,d^b=di,e^b=eh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89322);