#374385

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1712 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,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, 281581);