#419199

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60366	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1868 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326395);