#203611

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dhi,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, 110807);