#97792

order	512
nilpotency class	3
isoclinism	isoclinic class of #97718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

527 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4988);