#113111

order	512
nilpotency class	3
isoclinism	isoclinic class of #113110
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20307);