#493851

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1146 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)(5,6,9,15,10,16,25,33,11,12,23,27,8,14,22,31)(7,17,20,34,21,13,28,29,18,30,24,36,19,26,35,32);(6,12)(9,23)(13,26)(14,16)(17,30)(20,24)(22,25)(28,35);(2,4)(5,7)(6,13,12,26)(8,19)(9,24)(10,21)(11,18)(14,17,16,30)(15,32,27,29)(20,23)(22,28)(25,35)(31,36,33,34);(7,18)(13,26)(17,30)(19,21)(20,24)(28,35)(29,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292819, #72292820, #72292826, …
maximal normal subgroup of	#72292819, #72292820, #72292826, …
maximal quotient of	#72293289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401047);