#204453

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
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×#3169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111649);