#361483

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

1413 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,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, 268679);