#456880

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364076);