#162005

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

788 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25)(33,35,37,42)(34,38,39,41)(36,44,45,43)(40,48,47,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(35,42)(38,41)(43,44)(46,48);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(18,23)(22,31)(24,28)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69201);