#493595

order	512
nilpotency class	4
isoclinism	isoclinic class of #493594
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1790
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62874	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2442 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)(3,12,15,8)(4,9)(5,10,20,21)(6,11)(7,14)(13,28,16,24)(17,26,31,32)(18,27)(19,23)(22,29)(25,30);(1,3)(2,7)(4,16)(5,14)(6,15)(8,21)(9,25)(10,12)(11,23)(13,18)(17,30)(19,20)(22,27)(24,26)(28,32)(29,31);(1,3)(2,8)(4,13)(5,19)(6,15)(7,21)(9,24)(10,23)(11,12)(14,20)(16,18)(17,30)(22,32)(25,26)(27,28)(29,31);(1,4,6,18)(2,9,11,27)(3,13,15,16)(5,17,20,31)(7,22,23,25)(8,24,12,28)(10,26,21,32)(14,29,19,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234209, #72234212, #72261398, …
maximal normal subgroup of	#72234209, #72234212, #72261398, …
maximal quotient of	#72238622, #72320160

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400791);