#411653

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318849);