#498235

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(6,8)(9,10,13,16,15,20,12,18)(11,21,22,38,25,39,24,36)(14,28,27,37,29,35,26,19)(17,32,33,23,34,30,31,40);(1,2)(3,6)(4,5)(7,8)(10,16,20,18)(14,29)(17,31,34,33)(19,35,37,28)(21,38,39,36)(23,40)(26,27)(30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,22)(13,24)(14,30)(15,25)(16,31)(18,33)(19,36)(20,34)(21,35)(23,27)(26,40)(28,39)(29,32)(37,38);(2,5)(6,8)(9,12,15,13)(10,18,20,16)(11,22,25,24)(14,26,29,27)(17,33,34,31)(19,35,37,28)(21,38,39,36)(23,30,40,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=deh,f^b=fi,e^c=ei,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	#72313590, #72313596, #72313651, …
maximal normal subgroup of	#72313590, #72313596, #72313651, …
maximal quotient of	#72343322, #72343367, #72343389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405431);