#147760

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=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, 54956);