#458523

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

950 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 365719);