#147549

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1004 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54745);