#416209

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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)(7,15,17,11)(8,19)(9,23)(10,13)(12,21)(14,20)(16,29,31,33)(18,25,30,28)(22,34)(24,32,27,36)(26,35);(5,7,9,18,10,17,20,30)(6,11,23,28,13,15,14,25)(8,16,21,32,22,31,35,36)(12,24,34,29,26,27,19,33);(2,4)(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,27)(18,28)(19,21)(20,23)(22,26)(24,31)(25,30)(29,32)(33,36)(34,35);(5,8,20,35,10,22,9,21)(6,12,23,34,13,26,14,19)(7,16,30,36,17,31,18,32)(11,24,28,29,15,27,25,33)

Polycyclic presentation:

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

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, 323405);