#183639

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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

714 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90835);