#149769

order	512
nilpotency class	5
isoclinism	isoclinic class of #149769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11504
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

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

normal subgroups only

This group has 301 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=i,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=efi,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, 56965);