#100871

order	512
nilpotency class	3
isoclinism	isoclinic class of #100827
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	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 467 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,6,7,5,8,4,3)(9,10,13,16,14,11,12,15)(18,21,22,24);(1,3,6,8,5,7,4,2)(9,11,14,10)(12,15,13,16)(19,23)(21,24);(1,4,5,6)(2,7,8,3)(9,10,14,11)(12,15,13,16)(17,18)(19,24)(20,22)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8067);