#93311

order	512
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂³	#341
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,27,23,28,36,42,24,29,37,43,22,38,21,26)(19,30,31,25,34,39,46,40,35,45,47,41,33,48,32,44);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,21,32,22,33,37,47,24,35,36,46,23,34,20,31)(18,25,27,39,28,40,42,45,29,41,43,48,38,44,26,30)

Polycyclic presentation:

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

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, 507);