#147822

order	512
nilpotency class	4
isoclinism	isoclinic class of #129633
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

1115 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,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, 55018);