#497142

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#33719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,e^2=i,f^2=h,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	#72264753, #72264781, #72265409, …
maximal normal subgroup of	#72264753, #72264781, #72265409, …
maximal quotient of	#72343545, #72343574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404338);