#432006

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1182 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:

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

Polycyclic presentation:

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

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, 339202);