#149767

order	512
nilpotency class	5
isoclinism	isoclinic class of #149765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11503
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2920 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=ef,g^b=gi,e^c=ei,e^d=eh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56963);