#141576

order	512
nilpotency class	4
isoclinism	isoclinic class of #141425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,d^2=gh,e^2=ghi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48772);