#204478

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1028 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111674);