#107218

order	512
nilpotency class	3
isoclinism	isoclinic class of #107214
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	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 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:

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

Polycyclic presentation:

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

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, 14414);