#497054

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264923, #72264975, #72265409, …
maximal normal subgroup of	#72264923, #72264975, #72265409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404250);