#432002

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339198);