#238653

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145849);