#6440356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317836, #72318287, #72318334
maximal normal subgroup of	#72317836, #72318287, #72318334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347552);