#395179

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 302375);