#107633

order	512
nilpotency class	3
isoclinism	isoclinic class of #107630
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,5)(4,12)(6,9)(7,10)(8,14)(11,15)(13,16)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,30)(25,34,41,33)(29,44,46,38)(31,39)(37,40)(43,45,48,47);(1,3,5,10,7,9,6,2)(4,11,14,16,15,12,13,8)(17,19)(18,24,27,40)(20,29)(21,30)(22,37)(23,31)(25,38,41,44)(26,28,36,39)(32,48,42,43)(33,45)(34,47)(35,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,38)(26,42)(27,41)(28,43)(30,47)(31,46)(32,36)(37,45)(39,48)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14829);