#427306

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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)(5,6,11,8)(7,15,19,30)(9,13,24,20)(10,21,26,14)(12,25,16,18)(17,32,33,28)(22,31,35,27)(23,34,36,29);(5,7)(6,12)(8,16)(9,22)(10,25)(11,19)(13,28)(14,30)(15,21)(17,23)(18,26)(20,32)(24,35)(27,29)(31,34)(33,36);(5,8,11,6)(7,16,19,12)(9,20,24,13)(10,14,26,21)(15,25,30,18)(17,31,33,27)(22,28,35,32)(23,29,36,34);(2,4)(5,9)(6,13)(7,17,19,33)(8,20)(10,23)(11,24)(12,27,16,31)(14,29)(15,28,30,32)(18,22,25,35)(21,34)(26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334502);