#148212

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1802
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,e^c=eh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55408);