#496494

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

820 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)(3,7)(9,10,13,17,14,19,12,25,15,20,26,33,27,34,28,18)(11,16,23,29,24,32,22,38,21,30,36,39,37,40,35,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,29)(18,31)(19,32)(20,30)(25,38)(26,36)(27,37)(28,35)(33,39)(34,40);(2,5)(6,8)(10,17)(12,26)(13,28)(14,27)(16,29)(18,19)(20,33)(22,36)(23,35)(24,37)(25,34)(30,39)(31,32)(38,40);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(22,35)(23,36)(24,37)(29,39)(31,38)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343247, #72343250, #72343264
maximal normal subgroup of	#72343247, #72343250, #72343264
maximal quotient of	#72343726

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403690);