#163304

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

665 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,26)(20,23)(21,25)(22,30)(24,28)(27,31)(29,32)(33,34,36,40,37,39,43,46)(35,38,42,45,41,44,47,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30)(33,35)(34,38)(36,42)(37,41)(39,44)(40,45)(43,47)(46,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31)(35,41)(38,44)(42,47)(45,48);(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)(34,39)(38,44)(40,46)(45,48)

Polycyclic presentation:

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

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, 70500);