#108019

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,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, 15215);