#492252

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
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

468 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 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, 399448);