#456517

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1126 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)(7,10)(8,19)(9,24,27,18)(11,15)(12,16)(13,22)(14,26,32,28)(17,23)(20,29,34,36)(21,30)(25,33,35,31);(1,2)(3,4)(5,7,11,16,12,15,10,6)(8,17,22,13,23,19,21,30)(9,18,26,32,27,24,28,14)(20,33,25,29,34,31,35,36);(2,4)(5,8,12,23)(6,13,16,30)(7,17,15,19)(9,25,27,35)(10,22,11,21)(14,31,32,33)(18,29,24,36)(20,28,34,26);(5,9)(6,14)(7,18)(8,20)(10,28)(11,26)(12,27)(13,29)(15,24)(16,32)(17,33)(19,31)(21,35)(22,25)(23,34)(30,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 363713);