#429571

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,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, 336767);