#493498

order	512
nilpotency class	4
isoclinism	isoclinic class of #493492
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2237 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)(3,15,4,19)(5,11)(6,12,23,30)(7,13)(8,14)(9,17)(10,16,28,18)(20,27)(21,24,22,29)(25,31)(26,32);(1,3,8,4)(2,9,14,27)(5,16,7,18)(6,17,23,20)(10,24,28,29)(11,25,13,26)(12,15,30,19)(21,31,22,32);(1,4,8,3)(2,10,14,28)(5,16,7,18)(6,20,23,17)(9,24,27,29)(11,19,13,15)(12,26,30,25)(21,31,22,32);(1,5)(2,11)(3,16)(4,18)(6,21)(7,8)(9,25)(10,19)(12,24)(13,14)(15,28)(17,31)(20,32)(22,23)(26,27)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260522, #72260554, #72260564
maximal normal subgroup of	#72260522, #72260554, #72260564
maximal quotient of	#72261187, #72262883

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400694);