#115740

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=hi,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22936);