#184134

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91330);