#181103

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1335 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=hi,b^a=be,e^a=eh,c^b=ci,f^b=fh,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, 88299);