#492234

order	512
nilpotency class	4
isoclinism	isoclinic class of #492224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310014
maximal normal subgroup of	#72310014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399430);