#148679

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	#1797
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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:

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

Polycyclic presentation:

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

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, 55875);