#93907

order	512
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1345
commutator	C₂×M₁₆	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7103
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

379 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11)(3,14,16,45,19,47,50,59)(4,20,22,53)(6,28,30,41)(7,12,26,42)(8,13,27,43)(9,31,32,48,34,58,40,15)(10,35,37,23)(17,46,52,64,18,36,49,62)(21,33,54,61,51,44,55,60)(24,38,25,57)(29,39,56,63);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,36)(11,40)(12,44)(13,34)(14,38)(15,23)(20,47)(22,55)(24,51)(25,54)(26,52)(27,50)(28,33)(29,48)(30,49)(31,56)(32,43)(35,58)(37,62)(39,46)(41,60)(42,61)(45,57)(53,59)(63,64)

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1103);