#113202

order	512
nilpotency class	3
isoclinism	isoclinic class of #113192
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	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,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, 20398);