#492513

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

535 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309422
maximal normal subgroup of	#72309422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399709);