#93364

order	512
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 294 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fghi,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=ci,d^b=dg

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, 560);