#238577

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
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

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

normal subgroups only

This group has 890 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)(21,22,26,30)(23,27,33,43)(24,34,35,47)(25,37,40,29)(28,39,45,36)(31,48,49,52)(32,46,38,42)(41,51,44,50);(1,2)(3,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,26,33)(22,27,30,43)(24,31,35,49)(25,38,40,32)(28,44,45,41)(29,46,37,42)(34,48,47,52)(36,51,39,50);(1,3,4,2)(6,12)(7,15)(8,9)(13,14)(16,17)(18,19)(21,24)(22,28)(23,31)(25,39)(26,35)(27,41)(29,47)(30,45)(32,51)(33,49)(34,37)(36,40)(38,50)(42,52)(43,44)(46,48);(24,35)(28,45)(31,49)(34,47)(36,39)(41,44)(48,52)(50,51)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,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, 145773);