#149302

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
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

504 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^c=di,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, 56498);