#413201

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2406 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)(7,15)(8,12)(9,22)(10,13)(11,17)(14,23)(16,30)(18,29)(19,28)(20,35)(21,26)(24,34)(25,33)(27,31)(32,36);(5,7,9,17,10,18,23,33)(6,11,14,15,13,25,22,29)(8,19,20,34,21,16,31,32)(12,24,27,28,26,36,35,30);(5,8)(6,12)(7,16,18,19)(9,20)(10,21)(11,24,25,36)(13,26)(14,27)(15,28,29,30)(17,32,33,34)(22,35)(23,31);(1,2,3,4)(6,13)(11,25)(12,26)(14,22)(15,29)(24,36)(27,35)(28,30)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=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, 320397);