#181716

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88912);