#107185

order	512
nilpotency class	3
isoclinism	isoclinic class of #107174
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

705 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)(17,18,21,25,22,26,33,39)(19,27,30,41,29,23,34,36)(20,24,32,38,31,37,44,47)(28,35,43,46,42,40,45,48);(1,3,5,8,6,10,14,2)(4,9,13,16,12,15,11,7)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,35,37,40)(25,36,39,41)(31,42)(32,43)(33,34)(38,46,47,48)(44,45);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,38)(26,37)(27,40)(29,42)(30,43)(33,44)(34,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,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, 14381);