#184148

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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:

(5,6,9,13)(7,11,14,18)(8,10,16,12)(15,17,20,19)(21,22,24,28,25,26,31,33)(23,27,29,34,30,32,35,36);(1,2,4,3)(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(1,3,4,2)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(22,26)(27,32)(28,33)(34,36);(7,14)(11,18)(15,20)(17,19)(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=g,d^2=fh,e^2=h,f^2=i,b^a=be,c^a=ci,e^a=eh,e^b=eh,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, 91344);