#494642

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1308 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)(6,11,14,19)(8,12)(9,21)(13,22,27,32)(15,24,29,23)(17,25)(18,31)(20,26)(28,34,36,33)(30,35);(5,6,12,24,21,29,8,19,10,14,26,23,9,15,20,11)(7,13,25,34,31,36,17,32,16,27,35,33,18,28,30,22);(2,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,22)(12,25)(14,27)(15,28)(19,32)(20,30)(21,31)(23,33)(24,34)(26,35)(29,36);(7,16)(13,27)(17,30)(18,31)(22,32)(25,35)(28,36)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250443, #72250478, #72250485, …
maximal normal subgroup of	#72250443, #72250478, #72250485, …
maximal quotient of	#72319267, #72319916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401838);