#161877

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,27)(20,25)(21,31)(22,26)(23,28)(24,29)(30,32)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(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)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19)(18,24)(20,30)(21,28)(22,29)(23,25)(26,27)(31,32)(35,41)(38,44)(42,47)(45,48);(17,20,22,21)(18,25,26,31)(19,28,29,30)(23,24,32,27)

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,c^a=ch,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, 69073);