#149257

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56453);