#496379

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 306 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403575);