#165412

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72608);