#205572

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1246 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)(3,4)(5,8)(6,7)(9,10)(11,20,23,31)(12,24)(13,18)(14,29)(15,19)(16,22,32,28)(17,27)(21,30,35,40)(25,38)(26,33)(34,39,37,36);(1,3,4,2,6,8,5,7)(9,11,13,28,15,23,14,22)(10,16,18,31,19,32,29,20)(12,21,26,39,25,35,27,36)(17,30,24,37,33,40,38,34);(9,12,14,27,15,25,13,26)(10,17,18,24,19,33,29,38)(11,21,22,36,23,35,28,39)(16,30,31,37,32,40,20,34);(1,4,6,5)(2,3,7,8)(12,25)(17,33)(21,35)(24,38)(26,27)(30,40)(34,37)(36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=h,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di

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, 112768);