#146064

order	512
nilpotency class	4
isoclinism	isoclinic class of #146013
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#470
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

196 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53260);