#393028

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=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, 300224);