#146559

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53755);