#117124

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6)(7,13)(8,10)(9,17)(11,15)(12,16)(14,20)(18,19)(21,22)(23,33)(24,38,42,47)(25,31)(26,30)(27,32)(28,35)(29,40,48,41)(34,49,51,43)(36,45)(37,44)(39,46,50,52);(1,2,3,4)(6,10)(11,19)(12,17)(13,20)(21,23,25,35,27,37,26,36)(22,28,30,33,32,45,31,44)(24,39,40,34,42,50,41,51)(29,46,47,43,48,52,38,49);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)(21,24)(22,29)(23,34)(25,41)(26,40)(27,42)(28,43)(30,38)(31,47)(32,48)(33,46)(35,39)(36,50)(37,51)(44,52)(45,49)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

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, 24320);