#457129

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

1114 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 364325);