#164535

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

739 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30)(33,34,37,39,38,42,36,41)(35,43,46,48,44,40,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32)(34,39)(36,37)(40,47)(41,42)(43,48)(45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(23,31)(25,26)(27,32)(33,35)(34,40)(36,45)(37,46)(38,44)(39,47)(41,48)(42,43);(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)(35,44)(40,43)(45,46)(47,48)

Polycyclic presentation:

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

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, 71731);