#99418

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

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

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

normal subgroups only

This group has 298 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=g,d^2=g,e^2=i,f^2=ghi,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, 6614);