#116053

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#399
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 638 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)(21,22,26,31,27,32,25,30)(23,28,36,44,34,42,35,43)(24,29,39,47,38,46,40,48)(33,49,37,52,50,41,51,45);(1,2,3,4)(6,10)(11,19)(12,17)(13,20)(21,23)(22,28)(24,37,38,51)(25,35)(26,36)(27,34)(29,45,46,52)(30,43)(31,44)(32,42)(33,39,50,40)(41,47,49,48);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)(21,24)(22,29)(23,33)(25,40)(26,39)(27,38)(28,41)(30,48)(31,47)(32,46)(34,50)(35,51)(36,37)(42,49)(43,52)(44,45)

Polycyclic presentation:

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

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, 23249);