#425935

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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

959 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)(7,11)(8,18,20,28)(9,13)(10,14)(12,21,27,22)(15,30,19,34)(16,25)(17,26)(23,29)(24,31,35,32)(33,36);(5,7,9,16,10,17,23,33)(6,11,13,25,14,26,29,36)(8,19,21,32,20,15,22,31)(12,24,18,30,27,35,28,34);(5,8,9,22,10,20,23,21)(6,12,13,28,14,27,29,18)(7,15,16,32,17,19,33,31)(11,24,25,34,26,35,36,30);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,24)(16,25)(17,26)(18,21)(19,35)(20,27)(22,28)(23,29)(30,31)(32,34)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333131);