#163219

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
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

837 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cgh,d^a=di,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, 70415);