#493465

order	512
nilpotency class	4
isoclinism	isoclinic class of #493464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2265 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)(4,9)(5,10,18,7)(6,11)(8,15)(13,25)(14,19,28,22)(16,24)(17,23,29,20)(21,26)(27,30,32,31);(2,7)(4,16)(5,18)(8,19)(9,23)(10,11)(12,22)(13,26)(14,28)(20,24)(21,31)(25,30);(1,3,6,15)(2,8)(4,13,16,26)(5,14,18,28)(7,19)(9,21)(10,22)(11,12)(17,27,29,32)(20,30)(23,31)(24,25);(1,4,6,16)(2,9,11,24)(3,13,15,26)(5,17,18,29)(7,20,10,23)(8,21,12,25)(14,27,28,32)(19,30,22,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261720, #72261759, #72319990
maximal normal subgroup of	#72261720, #72261759, #72319990
maximal quotient of	#72320176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400661);