#149184

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(5,6,11,17,10,15,22,29,12,18,21,13,9,16,7,19)(8,23,24,35,25,14,30,32,26,33,36,31,27,34,20,28);(1,2,3,4)(5,7,9,21,12,22,10,11)(6,13,15,19,18,17,16,29)(8,24,25,30,26,36,27,20)(14,31,23,32,34,35,33,28);(5,8,10,25,12,26,9,27)(6,14,16,23,18,34,15,33)(7,20,11,24,22,30,21,36)(13,28,29,31,17,32,19,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56380);