#218882

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

633 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)(3,7)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,24)(20,22)(25,26,30,27,29,32,37,34)(28,31,36,33,35,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,19)(12,21)(13,15)(14,17)(16,20)(18,23)(22,24)(25,27,30,32,29,34,37,26)(28,33,36,38,35,39,40,31);(3,8)(5,7)(9,11,12,22,13,20,14,18)(10,16,21,23,15,19,17,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(2,6)(5,7)(9,12,13,14)(10,17,15,21)(11,18,20,22)(16,23,19,24)(28,35)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126078);