#393027

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	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,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, 300223);