#99602

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#386
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

604 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798);