#494844

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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)(5,6,9,15,8,14,20,31,10,13,23,24,21,12,28,30)(7,11,19,27,18,26,34,35,17,25,22,32,16,29,33,36);(5,7,9,22,8,18,20,33,10,17,23,19,21,16,28,34)(6,11,24,27,14,26,30,35,13,25,15,32,12,29,31,36);(2,4)(6,12,13,14)(7,16,17,18)(9,23)(11,25)(15,31,24,30)(19,34,22,33)(20,28)(26,29);(6,13)(11,25)(12,14)(15,24)(26,29)(27,32)(30,31)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239415, #72239422, #72239430, …
maximal normal subgroup of	#72239415, #72239422, #72239430, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402040);