#394677

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2123 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^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,f^c=fi,g^c=gi,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, 301873);