#421904

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1166 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi,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, 329100);