#119053

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,8,7,3,6,4)(9,10)(11,19,22,15)(12,23)(13,18)(14,17)(16,25)(20,30,38,37)(21,35,27,31)(24,32,39,40)(26,36)(28,33)(29,34);(1,3)(2,7)(4,5)(6,8)(9,11)(10,15)(12,24)(13,27)(14,22)(16,32)(17,19)(18,35)(20,26)(21,29)(23,40)(25,39)(28,38)(30,33)(31,34)(36,37);(1,4,5,3,7,8,6,2)(9,12)(10,16,17,23)(11,20)(13,28)(14,25)(15,30,19,37)(18,36,34,33)(21,39)(22,38)(24,27)(26,29)(31,40,35,32)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26249);