#495732

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250051, #72250058, #72250086
maximal normal subgroup of	#72250051, #72250058, #72250086
maximal quotient of	#72265512

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402928);