#101823

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

471 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=gh,e^2=ghi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

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, 9019);