#204374

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

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^c=di

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, 111570);