#396365

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

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

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, 303561);