#107171

order	512
nilpotency class	3
isoclinism	isoclinic class of #107154
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	D₄
LCS	G ▷ C₂²×C₄ ▷ 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:

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

Polycyclic presentation:

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

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, 14367);