#213523

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

918 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 218 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)(3,7)(4,6)(5,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,31)(28,38,40,32)(29,33)(30,34)(35,49,51,43)(36,44)(37,45)(39,52,54,46)(41,47)(42,48)(50,56,53,55);(1,3)(2,5)(4,8)(6,7)(25,27,30,37)(26,31,34,45)(28,35,40,51)(29,41,42,36)(32,43,38,49)(33,47,48,44)(39,53,54,50)(46,56,52,55);(3,8)(5,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,28,30,40)(26,32,34,38)(27,35,37,51)(29,39,42,54)(31,43,45,49)(33,46,48,52)(36,50,41,53)(44,55,47,56);(2,6)(5,7)(11,17)(14,20)(18,23)(21,24)(25,29,30,42)(26,33,34,48)(27,36,37,41)(28,39,40,54)(31,44,45,47)(32,46,38,52)(35,50,51,53)(43,55,49,56)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,d^c=di,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120719);