#107607

order	512
nilpotency class	3
isoclinism	isoclinic class of #107590
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14803);