#498470

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
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	#33580	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ei,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,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309518, #72310198, #72310208, …
maximal normal subgroup of	#72309518, #72310198, #72310208, …
maximal quotient of	#72343499, #72343594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405666);