#92918

order	512
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1277
prime radical	C₂³	#335
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

150 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fh,d^b=dhi

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, 114);