#413469

order	512
nilpotency class	3
isoclinism	isoclinic class of #399124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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:

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

Polycyclic presentation:

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

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, 320665);