#358641

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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)(7,18,23,34)(8,24)(9,15)(10,16)(11,27)(12,17)(13,21,30,22)(14,25)(19,29,36,28)(20,33,35,32)(26,31);(1,2)(3,4)(5,7,11,22,12,23,10,21)(6,13,16,18,17,30,27,34)(8,19,25,35,26,36,9,20)(14,28,24,32,15,29,31,33);(2,4)(5,8,11,25,12,26,10,9)(6,14,16,24,17,15,27,31)(7,19,22,35,23,36,21,20)(13,28,18,32,30,29,34,33);(5,9,10,26,12,25,11,8)(6,15,16,31,17,14,27,24)(7,20,21,36,23,35,22,19)(13,29,18,33,30,28,34,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265837);