#416212

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1362 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)(5,6,9,14,10,13,23,15)(7,16,19,27,21,11,20,17)(8,22,24,12,25,29,36,28)(18,32,33,26,35,31,34,30);(1,2,3,4)(5,7,9,20,10,21,23,19)(6,11,14,27,13,16,15,17)(8,18,24,34,25,35,36,33)(12,26,29,32,28,30,22,31);(5,6)(7,17)(8,12)(9,15)(10,13)(11,19)(14,23)(16,20)(18,31)(21,27)(22,24)(25,28)(26,33)(29,36)(30,34)(32,35);(5,8,23,36,10,25,9,24)(6,12,14,29,13,28,15,22)(7,18,19,33,21,35,20,34)(11,26,27,32,16,30,17,31)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,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, 323408);