#147633

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54829);