#161931

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

Polycyclic presentation:

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

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, 69127);