#394538

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	#35660	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2170 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)(7,13)(8,20)(9,12)(10,26)(11,15)(14,23)(16,33)(17,19)(18,30)(21,28)(22,24)(25,32)(27,29)(31,35)(34,36);(1,2)(3,4)(5,7)(6,13)(8,21)(9,17)(10,18)(11,27)(12,19)(14,31)(15,29)(16,24)(20,28)(22,33)(23,35)(25,36)(26,30)(32,34);(1,3,4,2)(5,8)(6,14)(7,16)(9,22)(10,24)(11,23)(12,25)(13,28)(15,32)(17,34)(18,36)(19,35)(20,26)(21,27)(29,31)(30,33);(5,9,12,6)(7,17,19,13)(8,22,25,14)(10,15,11,26)(16,34,35,28)(18,29,27,30)(20,24,32,23)(21,33,36,31)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 301734);