#94007

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄⋊C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

129 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,20,25,21,27,34,45,22,28,35,46,37,49,58,69,23,26,36,44,38,47,59,67,39,48,60,68,61,70,76,79)(19,29,31,50,32,51,55,71,33,52,56,72,57,73,75,80,30,24,40,41,53,42,62,64,54,43,63,65,74,66,77,78);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,24,26,29)(20,31)(21,32)(22,33)(23,30)(25,41,44,50)(27,42,47,51)(28,43,48,52)(34,55)(35,56)(36,40)(37,57)(38,53)(39,54)(45,64,67,71)(46,65,68,72)(49,66,70,73)(58,75)(59,62)(60,63)(61,74)(69,78,79,80)(76,77)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,e^a=efi,c^b=ci,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1203);