#465670

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1101 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:

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

Polycyclic presentation:

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

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, 372866);