#93807

order	512
nilpotency class	5
isoclinism	isoclinic class of #93784
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1347
commutator	C₂²×C₁₆	C₂×C₄
Frattini, cosocle	C₂×C₂²⋊C₁₆	C₂²
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₂²×C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂×C₂²⋊C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

145 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ghi,c^2=fh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi

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, 1003);