#6369756

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,f^a=fi,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316924, #72316937, #72316946
maximal normal subgroup of	#72316924, #72316937, #72316946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276952);