#470284

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

1215 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)(7,12)(8,13)(9,11)(10,15)(14,20)(16,27)(17,26)(18,31)(19,34)(21,25)(22,30)(23,24)(28,35)(29,33)(32,36);(6,11,15,14)(7,16)(12,24,27,31)(13,25,30,29)(17,32)(18,23)(19,35)(26,28,36,34);(5,7,10,16)(6,12,15,27)(8,19,22,35)(9,23,20,18)(11,24,14,31)(13,28,30,34)(17,33,32,21)(25,36,29,26);(2,4)(5,8,20,33,10,22,9,21)(6,13,14,29,15,30,11,25)(7,17,23,35,16,32,18,19)(12,26,24,28,27,36,31,34)

Polycyclic presentation:

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

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, 377480);