#147836

order	512
nilpotency class	4
isoclinism	isoclinic class of #129627
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×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1063 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,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, 55032);