#481718

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#392	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1035 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:

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

Polycyclic presentation:

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

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, 388914);