#394556

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2138 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:

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

Polycyclic presentation:

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

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, 301752);