#465673

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
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

1105 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)(6,12)(7,17)(8,22)(9,11)(13,27)(15,16)(18,29)(19,23)(24,25)(26,31)(30,36)(33,34);(1,2,4,3)(5,6,11,16,10,15,9,12)(7,18,21,29,17,27,20,13)(8,14,24,31,25,32,22,26)(19,30,23,35,33,36,34,28);(1,2)(3,4)(5,7,10,17)(6,13,15,29)(8,23,25,34)(9,21,11,20)(12,18,16,27)(14,30,32,36)(19,22,33,24)(26,35,31,28);(1,3,4,2)(5,8,10,25)(6,14,15,32)(7,19,17,33)(9,22,11,24)(12,26,16,31)(13,28,29,35)(18,30,27,36)(20,34,21,23)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,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, 372869);