#416206

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	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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)(7,16,19,12)(8,22)(9,15)(10,26)(11,14)(13,24)(17,21,28,20)(18,31,32,35)(23,29)(25,36)(27,34,30,33);(1,2,3,4)(5,7,10,21,11,19,9,20)(6,12,26,17,14,16,15,28)(8,18,24,34,25,32,23,33)(13,27,36,31,29,30,22,35);(5,6)(7,17)(8,13)(9,26)(10,15)(11,14)(12,21)(16,20)(18,31)(19,28)(22,24)(23,36)(25,29)(27,34)(30,33)(32,35);(5,8,9,23,11,25,10,24)(6,13,26,36,14,29,15,22)(7,18,20,33,19,32,21,34)(12,27,17,31,16,30,28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=hi,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,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, 323402);