#394632

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
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×#3008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2218 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:

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

Polycyclic presentation:

2^9|c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 301828);