#361482

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
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

1467 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)(5,6,10,16,11,13,9,15)(7,17,18,26,21,12,20,28)(8,22,23,30,24,14,25,29)(19,27,33,35,34,31,32,36);(1,2,4,3)(5,7)(6,12)(8,19)(9,18)(10,20)(11,21)(13,17)(14,27)(15,28)(16,26)(22,31)(23,32)(24,34)(25,33)(29,36)(30,35);(6,13)(7,18)(9,10)(12,26)(14,22)(17,28)(19,32)(20,21)(23,25)(27,35)(31,36)(33,34);(1,3,4,2)(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,27)(13,22)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(26,35)(28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=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, 268678);