#161712

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

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68908);