#493475

order	512
nilpotency class	4
isoclinism	isoclinic class of #493474
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62820	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3484 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^d=fi,g^e=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233535, #72258028, #72258042, …
maximal normal subgroup of	#72233535, #72258028, #72258042, …
maximal quotient of	#72233513, #72303345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400671);