#141821

order	512
nilpotency class	4
isoclinism	isoclinic class of #141643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286992, #72287173, #72287175
maximal normal subgroup of	#72286992, #72287173, #72287175

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49017);