#382622

order	512
nilpotency class	3
isoclinism	isoclinic class of #213788
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289818);