#99502

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#385
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

323 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6698);