#184222

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 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:

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

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=chi,d^a=di,e^a=eh,c^b=ci,e^b=eh

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, 91418);