#361313

order	512
nilpotency class	3
isoclinism	isoclinic class of #190034
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1532 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)(7,11,17,24)(8,18,20,12)(9,13,21,26)(10,14)(15,27)(16,30)(19,32)(22,29)(23,31)(25,33)(28,34,36,35);(1,2)(3,4)(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,22,18,27)(13,23,26,30)(19,28)(20,29)(21,31)(25,34,32,35)(33,36);(2,4)(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,25)(14,18)(16,28)(17,29)(21,33)(23,34)(24,27)(26,32)(30,35)(31,36);(5,9,10,21)(6,13)(7,16,17,31)(8,19,20,33)(11,23)(12,25)(14,26)(15,28,29,36)(18,32)(22,34)(24,30)(27,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fi,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,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, 268509);