#427881

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1258 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)(5,6,10,15)(7,16,20,28)(8,21,25,13)(9,12,26,14)(11,17,29,19)(18,31,34,35)(22,27,33,36)(23,32,24,30);(5,7)(6,11)(8,22)(9,17)(10,20)(12,16)(13,31)(14,28)(15,29)(18,23)(19,26)(21,35)(24,34)(25,33)(27,32)(30,36);(6,12)(7,17)(9,26)(13,30)(14,15)(16,28)(18,33)(19,20)(21,32)(22,34)(23,24)(31,35);(2,4)(5,8,10,25)(6,13,15,21)(7,18)(9,23,26,24)(11,27)(12,30,14,32)(16,31)(17,33)(19,22)(20,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 335077);