#391219

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298415);