#147832

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×#532	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1081 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=di,e^b=ehi,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, 55028);