#183641

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

742 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,d^a=dh,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90837);