#162373

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 69569);