#6431765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314709, #72314796, #72314807, …
maximal normal subgroup of	#72314709, #72314796, #72314807, …
maximal quotient of	#72316209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338961);