#93655

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=h,e^2=fgh,f^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,c^b=cg,d^b=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 851);