#493601

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	C₂×#41	C₂⁴
prime radical	#62874	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2428 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:

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

Polycyclic presentation:

2^9|a^2=g,g^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	#72340414, #72340644
maximal normal subgroup of	#72340414, #72340644
maximal quotient of	#72340419, #72340425

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400797);