#204416

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#447	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,11,14)(7,15,19,16)(8,9,13,10)(12,17,20,18)(21,22,25,29,26,30,34,36)(23,27,31,24,28,33,35,32);(1,2,3,4)(5,7,9,17,11,19,10,18)(6,12,8,16,14,20,13,15)(21,23,25,31,26,28,34,35)(22,27,29,24,30,33,36,32);(5,8,10,6,11,13,9,14)(7,16,18,12,19,15,17,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,11,10)(6,13,14,8)(7,17,19,18)(12,15,20,16)(23,28)(24,32)(27,33)(31,35)

Polycyclic presentation:

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

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, 111612);