#498625

order	512
nilpotency class	4
isoclinism	isoclinic class of #484139
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264905, #72264923, #72264984, …
maximal normal subgroup of	#72264905, #72264923, #72264984, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405821);