#123530

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30726);