#165624

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72820);