#318361

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225557);