#161999

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

776 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)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(33,35)(34,38)(36,44)(37,42)(39,41)(40,48)(43,45)(46,47);(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)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,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, 69195);