#205579

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112775);