#219505

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=ef,d^2=ei,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,g^c=gh

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, 126701);