#416375

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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)(7,16,8,12)(9,22)(10,25)(11,14)(13,23)(15,20)(17,31,21,35)(18,27,19,28)(24,36)(26,33,29,32)(30,34);(5,7,10,19,11,8,20,18)(6,12,25,27,14,16,15,28)(9,17,23,33,24,21,34,32)(13,26,36,31,30,29,22,35);(2,4)(5,8)(6,12,14,16)(7,11)(9,17)(10,18)(13,29,30,26)(15,28,25,27)(19,20)(21,24)(22,31,36,35)(23,33)(32,34);(5,9,10,23,11,24,20,34)(6,13,15,22,14,30,25,36)(7,17,19,33,8,21,18,32)(12,26,28,35,16,29,27,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=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, 323571);