#161688

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(35,41)(38,44)(42,47)(45,48);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239871, #72239873
maximal normal subgroup of	#72239871, #72239873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68884);