#394637

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
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	C₂×#3010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2164 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 301833);