#403985

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2601 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,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, 311181);