#126920

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#323
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,h^2=i,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34116);