#101567

order	512
nilpotency class	3
isoclinism	isoclinic class of #101519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 494 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8763);