#165000

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
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

688 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(18,22,24,27)(23,29,30,32)(33,34,37,41,38,42,36,40)(35,43,44,39,46,47,45,48);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10)(17,18,21,24)(19,23,25,30)(20,27,28,22)(26,32,31,29)(33,35,38,46)(34,39,42,48)(36,44,37,45)(40,47,41,43);(2,8)(7,15)(9,13)(10,16)(17,19,21,25)(18,23,24,30)(20,26,28,31)(22,29,27,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,48,43);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72196);