#147785

order	512
nilpotency class	4
isoclinism	isoclinic class of #129617
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54981);