#357897

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2546 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265093);