#107421

order	512
nilpotency class	3
isoclinism	isoclinic class of #107406
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

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14617);