#425931

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(7,12)(8,20,22,31)(9,14)(10,25)(11,15)(13,23,30,24)(16,29,32,36)(17,27)(18,34)(19,28)(21,26,33,35);(5,7,9,17,11,19,10,18)(6,12,14,27,15,28,25,34)(8,21,23,16,22,33,24,32)(13,29,20,35,30,36,31,26);(5,8,9,24,11,22,10,23)(6,13,14,31,15,30,25,20)(7,16,17,21,19,32,18,33)(12,26,27,36,28,35,34,29);(2,4)(5,6)(7,12)(8,13)(9,14)(10,25)(11,15)(16,26)(17,27)(18,34)(19,28)(20,23)(21,36)(22,30)(24,31)(29,33)(32,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^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=de,e^c=ei,f^c=fi,g^c=gi,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, 333127);