#147191

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=di,e^c=ei

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, 54387);