#100536

order	512
nilpotency class	3
isoclinism	isoclinic class of #100235
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 527 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,d^2=gh,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7732);