#395980

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 303176);