#141536

order	512
nilpotency class	4
isoclinism	isoclinic class of #141454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=ghi,e^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48732);