#511660

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,43)(36,45)(37,46)(38,44)(39,49)(40,48)(41,47)(42,50)(51,55)(52,54)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,12)(10,15,17,18)(13,20,21,19)(16,23,24,22)(25,27)(26,31)(28,35)(29,42)(30,38)(32,44)(33,43)(34,50)(36,39)(37,40)(41,52)(45,49)(46,48)(47,54)(51,53)(55,56);(2,6)(5,8)(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(27,35)(28,39)(29,40)(31,43)(33,49)(34,48)(36,37)(38,52)(41,53)(42,51)(44,54)(45,46)(47,56)(50,55);(3,7)(5,8)(9,13,14,21)(10,16,17,24)(11,19,12,20)(15,22,18,23)(26,32)(31,44)(33,47)(34,48)(43,54)(45,55)(46,50)(49,56)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

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, 418856);