#113069

order	512
nilpotency class	3
isoclinism	isoclinic class of #113067
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 20265);