#161767

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

795 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,27)(20,26)(21,24)(22,25)(23,28)(29,32)(30,31)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,27,31,32)(35,41)(38,44)(42,47)(45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,25)(23,31)(24,26)(27,32)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,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, 68963);