#397962

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1902 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,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,g^b=gi,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, 305158);